Systems of Equations and Matrices

Sstems of Equations and Matrices A sstem of equations is a collection of two or more variables In this chapter, ou should learn the following How to use the methods of substitution and elimination to solve sstems of linear equations in two variables () How to solve multivariable linear sstems () How to solve sstems of inequalities () How to use matrices to solve sstems of linear equations () How to perform operations with matrices () How to find inverses of matrices and use inverse matrices to solve sstems of linear equations () How to find determinants of square matrices (7) Graham Hewood / istockphotocom How can ou use a matri to model the number of people in the United States who participate in snowboarding? (See Section, Eercise ) The graphs above show the three possible tpes of solutions for a sstem of two linear equations in two variables: infinitel man solutions, no solution, and one solution (See Section ) 8

8 Chapter Sstems of Equations and Matrices Sstems of Linear Equations in Two Variables Use the method of substitution to solve sstems of equations in two variables Use the method of elimination to solve sstems of linear equations in two variables Interpret graphicall the numbers of solutions of sstems of linear equations in two variables Use sstems of linear equations in two variables to model and solve real-life problems The Method of Substitution Up to this point in the tet, most problems have involved either a function of one variable or a single equation in two variables However, man problems in science, business, and engineering involve two or more equations in two or more variables To solve such problems, ou need to find solutions of a sstem of equations Here is an eample of a sstem of two equations in two unknowns Equation Equation A solution of this sstem is an ordered pair that satisfies each equation in the sstem Finding the set of all solutions is called solving the sstem of equations For instance, the ordered pair, is a solution of this sstem To check this, ou can substitute for and for in each equation Check (, ) in Equation and Equation :?? Write Equation Substitute for and for Solution checks in Equation Write Equation Substitute for and for Solution checks in Equation In this chapter, ou will stud four was to solve sstems of equations, beginning with the method of substitution The guidelines for solving a sstem of equations b the method of substitution are summarized below GUIDELINES FOR SOLVING A SYSTEM OF EQUATIONS BY THE METHOD OF SUBSTITUTION Solve one of the equations for one variable in terms of the other Substitute the epression found in Step into the other equation to obtain an equation in one variable Solve the equation obtained in Step Back-substitute the value obtained in Step into the epression obtained in Step to find the value of the other variable Check that the solution satisfies each of the original equations

Sstems of Linear Equations in Two Variables 8 The term back-substitution implies that ou work backwards First ou solve for one of the variables, and then ou substitute that value back into one of the equations in the sstem to find the value of the other variable The back-substitution reduces the two-equation sstem to one equation in a single variable EXAMPLE Solving a Sstem of Equations b Substitution EXPLORATION Use a graphing utilit to graph and in the same viewing window Use the zoom and trace features to find the coordinates of the point of intersection What is the relationship between the point of intersection and the solution found in Eample? Solve the sstem of equations Equation Equation Solution Begin b solving for in Equation Solve for in Equation Net, substitute this epression for into Equation and solve the resulting single-variable equation for Write Equation Substitute for Distributive Propert Combine like terms Divide each side b Finall, ou can solve for b back-substituting into the equation, to obtain Write revised Equation Substitute for Solve for The solution is the ordered pair, You can check this solution as follows Check Substitute, into Equation : Write Equation? Substitute for and Solution checks in Equation Substitute, into Equation :? Write Equation Substitute for and Solution checks in Equation Because, satisfies both equations in the sstem, it is a solution of the sstem of equations STUDY TIP Because man steps are required to solve a sstem of equations, it is ver eas to make errors in arithmetic So, ou should alwas check our solution b substituting it into each equation in the original sstem

8 Chapter Sstems of Equations and Matrices The equations in Eample are linear The method of substitution can also be used to solve sstems in which one or both of the equations are nonlinear Such a sstem ma have more than one solution EXPLORATION Use a graphing utilit to graph the two equations in Eample 7 in the same viewing window How man solutions do ou think this sstem has? Repeat this eperiment for the equations in Eample How man solutions does this sstem have? Eplain our reasoning EXAMPLE Substitution: Two-Solution Case Solve the sstem of equations 7 Solution Begin b solving for in Equation to obtain Net, substitute this epression for into Equation and solve for 7 7 8 Equation Equation Substitute for in Equation Simplif Write in general form Factor, Solve for Back-substituting these values of to solve for the corresponding values of produces the solutions, and, Check these in the original sstem The sstem of equations in Eample has two solutions It is possible that a sstem has no solutions, as shown in Eample EXAMPLE Substitution: No Real-Solution Case Solve the sstem of equations Solution Begin b solving for in Equation to obtain Net, substitute this epression for into Equation and solve for ± ± Equation Equation Substitute for in Equation Simplif Quadratic Formula Simplif Because the discriminant is negative, the equation has no (real) solution So, the original sstem has no (real) solution

Sstems of Linear Equations in Two Variables 8 The Method of Elimination So far, ou have studied one method for solving a sstem of equations: substitution Now ou will stud the method of elimination The ke step in this method is to obtain, for one of the variables, coefficients that differ onl in sign so that adding the equations eliminates the variable 7 Equation Equation Add equations Note that b adding the two equations, ou eliminate the -terms and obtain a single equation in Solving this equation for produces, which ou can then back-substitute into one of the original equations to solve for EXAMPLE Solving a Sstem of Equations b Elimination Solve the sstem of linear equations Equation Equation Solution Because the coefficients of differ onl in sign, ou can eliminate the -terms b adding the two equations Write Equation Write Equation 8 Add equations Solve for B back-substituting into Equation, ou can solve for Write Equation Substitute for Simplif Solve for The solution is, Check this in the original sstem, as follows Check?? Substitute into Equation Equation checks Substitute into Equation Equation checks NOTE Although ou could use either the method of substitution or the method of elimination to solve the sstem in Eample, ou ma find that the method of elimination is more efficient

8 Chapter Sstems of Equations and Matrices EXAMPLE Solving a Sstem of Equations b Elimination STUDY TIP To obtain coefficients (for one of the variables) that differ onl in sign, ou often need to multipl one or both of the equations b suitabl chosen constants Solve the sstem of linear equations 7 Equation Equation Solution For this sstem, ou can obtain coefficients of the -terms that differ onl in sign b multipling Equation b 7 7 Write Equation Multipl Equation b Add equations So, ou can see that B back-substituting this value of into Equation, ou can solve for 7 7 Write Equation Substitute for Combine like terms Solve for The solution is, Check this in both equations in the original sstem Check? 7 7? Substitute into Equation Equation checks Substitute into Equation Equation checks In Eample, the two sstems of linear equations 7 and 7 are called equivalent sstems because the have precisel the same solution set The operations that can be performed on a sstem of linear equations to produce an equivalent sstem are () interchanging an two equations, () multipling an equation b a nonzero constant, and () adding a multiple of one equation to an other equation in the sstem GUIDELINES FOR SOLVING A SYSTEM OF EQUATIONS BY THE METHOD OF ELIMINATION Obtain coefficients for or that differ onl in sign b multipling all terms of one or both equations b suitabl chosen constants Add the equations to eliminate one variable and solve the resulting equation Back-substitute the value obtained in Step into either of the original equations and solve for the other variable Check that the solution satisfies each of the original equations

Sstems of Linear Equations in Two Variables 87 Graphical Interpretation of Solutions It is possible for a general sstem of equations to have eactl one solution, two or more solutions, or no solution If a sstem of linear equations has two different solutions, it must have an infinite number of solutions GRAPHICAL INTERPRETATIONS OF SOLUTIONS For a sstem of two linear equations in two variables, the number of solutions is one of the following Number of Solutions Graphical Interpretation Slopes of Lines Eactl one solution The two lines intersect at one point The slopes of the two lines are not equal Infinitel man solutions The two lines coincide (are identical) The slopes of the two lines are equal No solution The two lines are parallel The slopes of the two lines are equal A sstem of linear equations is consistent if it has at least one solution A consistent sstem with eactl one solution is independent, whereas a consistent sstem with infinitel man solutions is dependent A sstem is inconsistent if it has no solution EXAMPLE Recognizing Graphs of Linear Sstems Match each sstem of linear equations with its graph Describe the number of solutions and state whether the sstem is consistent or inconsistent a b c i ii iii Solution a The graph of sstem (a) is a pair of parallel lines (ii) The lines have no point of intersection, so the sstem has no solution The sstem is inconsistent b The graph of sstem (b) is a pair of intersecting lines (iii) The lines have one point of intersection, so the sstem has eactl one solution The sstem is consistent c The graph of sstem (c) is a pair of lines that coincide (i) The lines have infinitel man points of intersection, so the sstem has infinitel man solutions The sstem is consistent STUDY TIP A comparison of the slopes of two lines gives useful information about the number of solutions of the corresponding sstem of equations To solve a sstem of equations graphicall, it helps to begin b writing the equations in slope-intercept form Tr doing this for the sstems in Eample

88 Chapter Sstems of Equations and Matrices In Eamples 7 and 8, note how ou can use the method of elimination to determine that a sstem of linear equations has no solution or infinitel man solutions EXAMPLE 7 No-Solution Case: Method of Elimination + = = Figure Solve the sstem of linear equations Equation Equation Solution To obtain coefficients that differ onl in sign, ou can multipl Equation b 7 Multipl Equation b Write Equation False statement Because there are no values of and for which 7, ou can conclude that the sstem is inconsistent and has no solution The lines corresponding to the two equations in this sstem are shown in Figure Note that the two lines are parallel and therefore have no point of intersection In Eample 7, note that the occurrence of a false statement, such as 7, indicates that the sstem has no solution In the net eample, note that the occurrence of a statement that is true for all values of the variables, such as, indicates that the sstem has infinitel man solutions EXAMPLE 8 Man-Solution Case: Method of Elimination Solve the sstem of linear equations (, ) (, ) = Equation Equation Solution To obtain coefficients that differ onl in sign, ou can multipl Equation b Multipl Equation b Write Equation Add equations Because the two equations are equivalent (have the same solution set), ou can conclude that the sstem has infinitel man solutions The solution set consists of all points, ling on the line Figure as shown in Figure Letting a, where a is an real number, ou can see that the solutions of the sstem are a, a In Eample 8, choose some values of a to find solutions of the sstem: for eample, if a, the solution is,, and if a, the solution is, Then check these solutions in the original sstem

Sstems of Linear Equations in Two Variables 89 Applications At this point, ou ma be asking the question How can I tell which application problems can be solved using a sstem of linear equations? The answer comes from the following considerations Does the problem involve more than one unknown quantit? Are there two (or more) equations or conditions to be satisfied? If one or both of these situations occur, the appropriate mathematical model for the problem ma be a sstem of linear equations Original flight WIND r r Return flight r + r Figure WIND EXAMPLE 9 An Application of a Linear Sstem An airplane fling into a headwind travels the -mile fling distance between Chicopee, Massachusetts and Salt Lake Cit, Utah in hours and minutes On the return flight, the same distance is traveled in hours Find the airspeed of the plane and the speed of the wind, assuming that both remain constant Solution The two unknown quantities are the speeds of the wind and the plane If r is the speed of the plane and r is the speed of the wind, then r r speed of the plane against the wind r r speed of the plane with the wind as shown in Figure Using the formula distance ratetime for these two speeds, ou obtain the following equations r r r r These two equations simplif as follows r r r r To solve this sstem b elimination, multipl Equation b Equation Equation r r r r r r r r Write Equation Multipl Equation b So, and r, 777 miles per hour r 7 miles per hour, r Check this solution in the original statement of the problem Add equations Speed of plane Speed of wind

87 Chapter Sstems of Equations and Matrices In a free market, the demands for man products are related to the prices of the products As the prices decrease, the demands b consumers increase and the amounts that producers are able or willing to suppl decrease Price per unit (in dollars) p (,,, ) Demand Suppl 7,,,, Number of units Figure EXAMPLE Finding the Equilibrium Point The demand and suppl equations for a new tpe of personal digital assistant are p Demand equation p Suppl equation where p is the price in dollars and represents the number of units Find the equilibrium point for this market The equilibrium point is the price p and number of units that satisf both the demand and suppl equations Solution Because p is written in terms of, begin b substituting the value of p given in the suppl equation into the demand equation p 9,, Write demand equation Substitute for p Combine like terms Solve for So, the equilibrium point occurs when the demand and suppl are each million units (See Figure ) The price that corresponds to this -value is obtained b back-substituting,, into either of the original equations For instance, back-substituting into the demand equation produces p,, $ The solution is,,, Check this in both equations in the original sstem Eercises See wwwcalcchatcom for worked-out solutions to odd-numbered eercises In Eercises 8, fill in the blanks A set of two or more equations in two or more variables is called a of A of a sstem of equations is an ordered pair that satisfies each equation in the sstem Finding the set of all solutions to a sstem of equations is called the sstem of equations The first step in solving a sstem of equations b the method of is to solve one of the equations for one variable in terms of the other variable The first step in solving a sstem of equations b the method of is to obtain coefficients for (or ) that differ onl in sign Two sstems of equations that have the same solution set are called sstems 7 A sstem of linear equations that has at least one solution is called, whereas a sstem of linear equations that has no solution is called 8 In business applications, the is defined as the price p and the number of units that satisf both the demand and suppl equations In Eercises 9, determine whether each ordered pair is a solution of the sstem of equations 9 (a), (b), 7 8 9 (c) (d),, (a), (b), 9 (c) (d) 7,, 7 e (a), (b), 7 (c), (d), (a) 9, 7 9 (b), log 9 8 9 (c), (d),

Sstems of Linear Equations in Two Variables 87 In Eercises 8, solve the sstem b the method of substitution 7 8 8 9 8 7 7 8 In Eercises 9, solve the sstem b the method of elimination Label each line with its equation To print an enlarged cop of the graph, go to the website wwwmathgraphscom 9 7 9 In Eercises 7, solve the sstem b the method of elimination and check an solutions analticall 7 8 9 u v u v 8 9 9 7 8 9 78 87 9 b m b m 8 r s r s 9 8 9 8 9 7 7 7 8 8 8 8

87 Chapter Sstems of Equations and Matrices In Eercises 7 7, use a graphing utilit to graph the lines in the sstem Use the graphs to determine if the sstem is consistent or inconsistent If the sstem is consistent, determine the number of solutions Then solve the sstem if possible 7 8 9 9 7 9 7 7 8 9 7 8 8 9 9 7 9 8 8 7 7 7 9 8 8 7 In Eercises 7 78, use an method to solve the sstem 7 7 7 7 9 7 7 7 7 7 7 7 77 78 9 7 8 WRITING ABOUT CONCEPTS 79 What is meant b a solution of a sstem of equations in two variables? 8 When solving a sstem of equations b substitution, how do ou recognize that the sstem has no solution? 8 Write a brief paragraph describing an advantages of substitution over the graphical method of solving a sstem of equations 8 Find equations of lines whose graphs intersect the graph of the parabola at (a) two points, (b) one point, and (c) no points (There is more than one correct answer) Use graphs to support our answer WRITING ABOUT CONCEPTS (continued) In Eercises 8 and 8, the graphs of the two equations appear to be parallel Yet, when the sstem is solved analticall, ou find that the sstem does have a solution Find the solution and eplain wh it does not appear on the portion of the graph that is shown 8 8 99 98 8 Briefl eplain whether or not it is possible for a consistent sstem of linear equations to have eactl two solutions 8 Give eamples of a sstem of linear equations that has (a) no solution and (b) an infinite number of solutions 87 Consider the sstem of equations a b c d e f (a) Find values for a, b, c, d, e, and f so that the sstem has one distinct solution (There is more than one correct answer) (b) Eplain how to solve the sstem in part (a) b the method of substitution and graphicall (c) Write a brief paragraph describing an advantages of the method of substitution over the graphical method of solving a sstem of equations CAPSTONE 88 Rewrite each sstem of equations in slope-intercept form and sketch the graph of each sstem What is the relationship among the slopes of the two lines, the number of points of intersection, and the number of solutions? (a) (b) 8 (c) 8

Sstems of Linear Equations in Two Variables 87 Suppl and Demand In Eercises 89 9, find the equilibrium point of the demand and suppl equations The equilibrium point is the price p and number of units that satisf both the demand and suppl equations Demand 89 p 9 p 9 p 9 p 9 DVD Rentals The weekl rentals for a newl released DVD of an animated film at a local video store decreased each week At the same time, the weekl rentals for a newl released DVD of a horror film increased each week Models that approimate the weekl rentals R for each DVD are R R 8 Suppl p 8 p p 8 p Animated film Horror film where represents the number of weeks each DVD was in the store, with corresponding to the first week (a) After how man weeks will the rentals for the two movies be equal? (b) Use a table to solve the sstem of equations numericall Compare our result with that of part (a) 9 Suppl and Demand The suppl and demand curves for a business dealing with wheat are Suppl: p Demand: p 88 7 where p is the price in dollars per bushel and is the quantit in bushels per da Use a graphing utilit to graph the suppl and demand equations and find the market equilibrium (The market equilibrium is the point of intersection of the graphs for > ) 9 Choice of Two Jobs You are offered two jobs selling dental supplies One compan offers a straight commission of % of sales The other compan offers a salar of $ per week plus % of sales How much would ou have to sell in a week in order to make the straight commission offer better? 9 Choice of Two Jobs You are offered two different jobs selling college tetbooks One compan offers an annual salar of $, plus a ear-end bonus of % of our total sales The other compan offers an annual salar of $, plus a ear-end bonus of % of our total sales Determine the annual sales required to make the second offer better 97 Investment Portfolio A total of $, is invested in two funds paing % and 8% simple interest (The % investment has a lower risk) The investor wants a earl interest income of $ from the two investments (a) Write a sstem of equations in which one equation represents the total amount invested and the other equation represents the $ required in interest Let and represent the amounts invested at % and 8%, respectivel (b) Use a graphing utilit to graph the two equations in the same viewing window As the amount invested at % increases, how does the amount invested at 8% change? How does the amount of interest income change? Eplain (c) What amount should be invested at % to meet the requirement of $ per ear in interest? 98 Investment Portfolio A total of $, is invested in two corporate bonds that pa % and % simple interest The investor wants an annual interest income of $9 from the investments What amount should be invested in the % bond? 99 Acid Miture Thirt liters of a % acid solution is obtained b miing a % solution with a % solution (a) Write a sstem of equations in which one equation represents the amount of final miture required and the other represents the percent of acid in the final miture Let and represent the amounts of the % and % solutions, respectivel (b) Use a graphing utilit to graph the two equations in part (a) in the same viewing window As the amount of the % solution increases, how does the amount of the % solution change? (c) How much of each solution is required to obtain the specified concentration of the final miture? Fuel Miture Five hundred gallons of 89-octane gasoline is obtained b miing 87-octane gasoline with 9-octane gasoline (a) Write a sstem of equations in which one equation represents the amount of final miture required and the other represents the amounts of 87- and 9-octane gasolines in the final miture Let and represent the numbers of gallons of 87-octane and 9-octane gasolines, respectivel (b) Use a graphing utilit to graph the two equations in part (a) in the same viewing window As the amount of 87-octane gasoline increases, how does the amount of 9-octane gasoline change? (c) How much of each tpe of gasoline is required to obtain the gallons of 89-octane gasoline?

87 Chapter Sstems of Equations and Matrices Geometr What are the dimensions of a rectangular tract of land if its perimeter is kilometers and its area is square kilometers? Geometr What are the dimensions of an isosceles right triangle with a two-inch hpotenuse and an area of square inch? Airplane Speed An airplane fling into a headwind travels the 8-mile fling distance between Pittsburgh, Pennslvania and Phoeni, Arizona in hours and minutes On the return flight, the distance is traveled in hours Find the airspeed of the plane and the speed of the wind, assuming that both remain constant Airplane Speed Two planes start from Los Angeles International Airport and fl in opposite directions The second plane starts hour after the first plane, but its speed is 8 kilometers per hour faster Find the airspeed of each plane if hours after the first plane departs the planes are kilometers apart Nutrition Two cheeseburgers and one small order of French fries from a fast-food restaurant contain a total of 8 calories Three cheeseburgers and two small orders of French fries contain a total of calories Find the caloric content of each item Nutrition One eight-ounce glass of apple juice and one eight-ounce glass of orange juice contain a total of 77 milligrams of vitamin C Two eight-ounce glasses of apple juice and three eight-ounce glasses of orange juice contain a total of 7 milligrams of vitamin C How much vitamin C is in an eight-ounce glass of each tpe of juice? 7 Prescriptions The numbers of prescriptions P (in thousands) filled at two pharmacies from through are shown in the table Year 7 8 9 Pharmac A 9 9 Pharmac B 8 (a) Use a graphing utilit to create a scatter plot of the data for pharmac A and use the regression feature to find a linear model Let t represent the ear, with t corresponding to Repeat the procedure for pharmac B (b) Assuming the numbers for the given five ears are representative of future ears, will the number of prescriptions filled at pharmac A ever eceed the number of prescriptions filled at pharmac B? If so, when? 8 Data Analsis A store manager wants to know the demand for a product as a function of the price The dail sales for different prices of the product are shown in the table Price, $ $ $ Demand, 7 (a) Find the least squares regression line for the data b solving the sstem for b 7a 7b 9a 9 a b a and b (b) Use the regression feature of a graphing utilit to confirm the result in part (a) (c) Use the graphing utilit to plot the data and graph the linear model from part (a) in the same viewing window (d) Use the linear model from part (a) to predict the demand when the price is $7 In Eercises 9, find a sstem of linear equations that has the given solution (There is more than one correct answer) 9, 8,,, In Eercises and, find the value of k such that the sstem of linear equations is inconsistent 8 k k 9 True or False? In Eercises, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false In order to solve a sstem of equations b substitution, ou must alwas solve for in one of the two equations and then back-substitute If a sstem consists of a parabola and a circle, then the sstem can have at most two solutions 7 If two lines do not have eactl one point of intersection, then the must be parallel 8 Solving a sstem of equations graphicall will alwas give an eact solution 9 If a sstem of linear equations has no solution, then the lines must be parallel To solve a sstem using the method of elimination, the equations in the sstem must be linear

Multivariable Linear Sstems 87 Multivariable Linear Sstems Use back-substitution to solve linear sstems in row-echelon form Use Gaussian elimination to solve sstems of linear equations Solve nonsquare sstems of linear equations Use sstems of linear equations in three or more variables to model and solve real-life problems Row-Echelon Form and Back-Substitution The method of elimination can be applied to a sstem of linear equations in more than two variables In fact, this method easil adapts to computer use for solving linear sstems with dozens of variables When elimination is used to solve a sstem of linear equations, the goal is to rewrite the sstem in a form to which back-substitution can be applied To see how this works, consider the following two sstems of linear equations Sstem of Three Linear Equations in Three Variables: (See Eample ) z 9 z 7 Equivalent Sstem in Row-Echelon Form: (See Eample ) z 9 z z The second sstem is said to be in row-echelon form, which means that it has a stair-step pattern with leading coefficients of After comparing the two sstems, it should be clear that it is easier to solve the sstem in row-echelon form, using back-substitution EXAMPLE Using Back-Substitution in Row-Echelon Form Solve the sstem of linear equations z 9 Equation z Equation z Equation Solution From Equation, ou know the value of z To solve for, substitute z into Equation to obtain Substitute for z Solve for Then substitute and z into Equation to obtain 9 Substitute for and for z Solve for The solution is,, and z, which can be written as the ordered triple,, Check this in all three equations in the original sstem of equations

87 Chapter Sstems of Equations and Matrices Christopher Lui/China Stock CHUI-CHANG SUAN-SHU One of the most influential Chinese mathematics books was the Chui-chang suan-shu or Nine Chapters on the Mathematical Art (written in approimatel BC) Chapter Eight of the Nine Chapters contained solutions of sstems of linear equations using positive and negative numbers One such sstem was as follows z 9 z This sstem was solved using column operations on a matri Matrices (plural for matri) will be discussed later in this chapter Gaussian Elimination Two sstems of equations are equivalent if the have the same solution set To solve a sstem that is not in row-echelon form, first convert it to an equivalent sstem that is in row-echelon form b using the following operations OPERATIONS THAT PRODUCE EQUIVALENT SYSTEMS Each of the following row operations on a sstem of linear equations produces an equivalent sstem of linear equations Interchange two equations Multipl one of the equations b a nonzero constant Add a multiple of one of the equations to another equation to replace the latter equation To see how this is done, take another look at the method of elimination, as applied to a sstem of two linear equations EXAMPLE Using Gaussian Elimination to Solve a Sstem Solve the sstem of linear equations Equation Equation Solution There are two strategies that seem reasonable: eliminate the variable or eliminate the variable The following steps show how to use the first strateg Interchange the two equations in the sstem Multipl the first equation b Add the multiple of the first equation to the second equation to obtain a new second equation New sstem in row-echelon form Notice in the first step that interchanging rows is an eas wa of obtaining a leading coefficient of Now back-substitute into Equation and solve for Substitute for Solve for The solution is and which can be written as the ordered pair,

Multivariable Linear Sstems 877 Rewriting a sstem of linear equations in row-echelon form usuall involves a chain of equivalent sstems, each of which is obtained b using one of the three basic row operations listed on the previous page This process is called Gaussian elimination, after the German mathematician Carl Friedrich Gauss (777 8) EXAMPLE Using Gaussian Elimination to Solve a Sstem STUDY TIP Arithmetic errors are often made when performing elementar row operations You should note the operation performed in each step so that ou can go back and check our work Solve the sstem of linear equations Equation Equation z 9 z 7 Equation Solution Because the leading coefficient of the first equation is, ou can begin b saving the at the upper left and eliminating the other -terms from the first column z 9 z z 9 z z 7 z 8 z 7 z z 9 z z Write Equation Write Equation Add Equation to Equation Adding the first equation to the second equation produces a new second equation Multipl Equation b Write Equation Add revised Equation to Equation Adding times the first equation to the third equation produces a new third equation Now that all but the first have been eliminated from the first column, go to work on the second column (You need to eliminate from the third equation) z 9 Adding the second equation to the z third equation produces a new z third equation Finall, ou need a coefficient of for z in the third equation z 9 Multipling the third equation b z produces a new third equation z This is the same sstem that was solved in Eample, and, as in that eample, ou can conclude that the solution is,, and z

878 Chapter Sstems of Equations and Matrices The net eample involves an inconsistent sstem one that has no solution The ke to recognizing an inconsistent sstem is that at some stage in the elimination process ou obtain a false statement such as EXAMPLE An Inconsistent Sstem (a) Solution: one point (b) Solution: one line (c) Solution: one plane Solve the sstem of linear equations Equation Equation z z z Equation Solution z z z z z z z z Adding times the first equation to the second equation produces a new second equation Adding times the first equation to the third equation produces a new third equation Adding times the second equation to the third equation produces a new third equation Because is a false statement, ou can conclude that this sstem is inconsistent and has no solution Moreover, because this sstem is equivalent to the original sstem, ou can conclude that the original sstem also has no solution As with a sstem of linear equations in two variables, the solution(s) of a sstem of linear equations in more than two variables must fall into one of three categories (d) Solution: none THE NUMBER OF SOLUTIONS OF A LINEAR SYSTEM For a sstem of linear equations, eactl one of the following is true There is eactl one solution There are infinitel man solutions There is no solution (e) Solution: none Figure In Section, ou learned that a sstem of two linear equations in two variables can be represented graphicall as a pair of lines that are intersecting, coincident, or parallel A sstem of three linear equations in three variables has a similar graphical representation it can be represented as three planes in space that intersect in one point (eactl one solution) [see Figure (a)], intersect in a line or a plane (infinitel man solutions) [see Figures (b) and (c)], or have no points common to all three planes (no solution) [see Figures (d) and (e)]

Multivariable Linear Sstems 879 EXAMPLE A Sstem with Infinitel Man Solutions Solve the sstem of linear equations Equation Equation z z Equation STUDY TIP In Eample, and are solved in terms of the third variable z To write the correct form of the solution to the sstem that does not use an of the three variables of the sstem, let a represent an real number and let Then solve for and The solution can then be written in terms of a, which is not one of the variables of the sstem z a Solution z z z z z Adding the first equation to the third equation produces a new third equation Adding times the second equation to the third equation produces a new third equation This result means that Equation depends on Equations and in the sense that it gives no additional information about the variables Because is a true statement, ou can conclude that this sstem will have infinitel man solutions However, it is incorrect to sa simpl that the solution is infinite You must also specif the correct form of the solution So, the original sstem is equivalent to the sstem z z In the last equation, solve for in terms of z to obtain z Back-substituting z in the first equation produces z Finall, letting z a, where a is a real number, the solutions to the given sstem are all of the form a, a, and z a So, ever ordered triple of the form a, a, a a is a real number is a solution of the sstem STUDY TIP When comparing descriptions of an infinite solution set, keep in mind that there is more than one wa to describe the set In Eample, there are other was to write the same infinite set of solutions For instance, b solving for and z in terms of and letting b (where b is a real number), the solutions could have been written as b, b, b b is a real number To convince ourself that this description produces the same set of solutions, consider the following Substitution Solution a b a b a b,,,,,,,,,,,,,,,,,,,,,,,, Same Solution Same Solution Same Solution

88 Chapter Sstems of Equations and Matrices Nonsquare Sstems So far, each sstem of linear equations ou have looked at has been square, which means that the number of equations is equal to the number of variables In a nonsquare sstem, the number of equations differs from the number of variables A sstem of linear equations cannot have a unique solution unless there are at least as man equations as there are variables in the sstem EXAMPLE A Sstem with Fewer Equations than Variables Solve the sstem of linear equations Equation z z Equation Solution Begin b rewriting the sstem in row-echelon form z z Adding times the first equation to the second equation produces a new second equation z z Multipling the second equation b produces a new second equation Solve for in terms of z to obtain z, and back-substitute z into Equation to obtain z Finall, b letting z a, where a is a real number, ou have the solution a, a, and z a So, ever ordered triple of the form a, a, a, where a is a real number, is a solution of the sstem Because there were originall three variables and onl two equations, the sstem cannot have a unique solution Applications (, ) Figure = (, ) (, ) EXAMPLE 7 Data Analsis: Curve-Fitting Find the equation of the parabola a b c whose graph passes through the points,,,, and, Solution Because the graph of a b c passes through the points,,,, and,, ou can substitute for and in the equation a b c for each ordered pair to produce the following sstem of linear equations a b c Equation : Substitute for and for a b c Equation : Substitute for and for a b c Equation : Substitute for and for The solution of this sstem is a, b, and c So, the equation of the parabola is, as shown in Figure

Multivariable Linear Sstems 88 s t = t = t = t = Figure 7 Recall that the height at time t of an object that is moving in a (vertical) line with constant acceleration g is given b the position function st gt v t s where st is the height of the object at time t, v is the initial velocit (at t ), and s is the initial height of the object Eample 8 demonstrates how a sstem of equations can be used to find the position function given the heights at various times EXAMPLE 8 Vertical Motion An object moving verticall is at the following heights at the specified times At t second, s feet At t seconds, s feet At t seconds, s feet Find the values of g, v, and s in the position function st gt v t s (See Figure 7) Solution B substituting t and s into the position function, ou can obtain three linear equations in g, v, and s When t : When t : When t : g v s g v s g v s g v s g v s 9g v s Solving this sstem ields g, v 8, and s, which can be written as, 8, This solution results in a position function of st t 8t and implies that the object was thrown upward at a velocit of 8 feet per second from a height of feet EXAMPLE 9 Investment Analsis An inheritance of $, was invested among three funds: a mone-market fund that paid % annuall, municipal bonds that paid % annuall, and mutual funds that paid 7% annuall The amount invested in mutual funds was $ more than the amount invested in municipal bonds The total interest earned during the first ear was $7 How much was invested in each tpe of fund? Solution Let,, and z represent the amounts invested in the mone-market fund, municipal bonds, and mutual funds, respectivel From the given information, ou can write the following equations z, Equation z Equation 7z 7 Equation Rewriting this sstem in standard form without decimals produces the following Equation Equation z, z, 7z 7, Equation Using Gaussian elimination to solve this sstem ields,, and z 7 So, $ was invested in the mone-market fund, $ was invested in municipal bonds, and $7 was invested in mutual funds

88 Chapter Sstems of Equations and Matrices Eercises See wwwcalcchatcom for worked-out solutions to odd-numbered eercises In Eercises, fill in the blanks A sstem of equations that is in form has a stair-step pattern with leading coefficients of A solution to a sstem of three linear equations in three unknowns can be written as an, which has the form,, z The process used to write a sstem of linear equations in row-echelon form is called elimination Interchanging two equations of a sstem of linear equations is a that produces an equivalent sstem A sstem of equations is called if the number of equations differs from the number of variables in the sstem The function st) gt v t s is called the function, and it models the height st of an object at time t that is moving in a vertical line with a constant acceleration g In Eercises 7, determine whether each ordered triple is a solution of the sstem of equations 7 z z 9 z (a),, (b),, (c),, (d),, 8 z 7 z 7z (a),, (b),, (c),, (d),, 9 z 8 z 7 (a) (b) (c),, (d) 8z z 7 (a) (c),, 7,, 8,, 9 (b) (d),,,,,,,, In Eercises, use back-substitution to solve the sstem of linear equations z z 8 z z z z 8 z z z z 8 z z In Eercises 7, solve the sstem of linear equations and check an solution analticall 7 8 z z z 7 z 9 z z 9 z z z 9 z z 7 z 9 z z z 7 z 7 7z z 9 z z 8z 9 z 8z z z z z z z z z 7 z z 7 z z 8z 8 8z 8 z z z 8 9 z z z z 8 z 7 z z 8 z z z 8

Multivariable Linear Sstems 88 z 9 7 7 8 z z z 9 z 8 z z z z w w w w z z z z w w w z 7 8 z z z z 7z z 9z z z WRITING ABOUT CONCEPTS In Eercises and, perform the row operation and write the equivalent sstem Add Equation to Equation Equation Equation z z z Equation What did this operation accomplish? Add times Equation to Equation Equation Equation z z z Equation What did this operation accomplish? Are the following two sstems of equations equivalent? Give reasons for our answer z z z z 7 z 7 z One of the following sstems is inconsistent and the other has one solution How can ou identif each b observation? 8 9 WRITING ABOUT CONCEPTS (continued) 7 When using Gaussian elimination to solve a sstem of linear equations, how can ou recognize that the sstem has no solution? Give an eample that illustrates our answer 8 Eplain the graphical significance of a sstem of three equations with three unknowns having a unique solution Vertical Motion In Eercises 9, an object moving verticall is at the given heights at the specified times Find the position function st gt v t s for the object 9 At t second, s 8 feet At t seconds, s 8 feet At t seconds, s feet At t second, s feet At t seconds, s feet At t seconds, s feet At t second, s feet At t seconds, s 7 feet At t seconds, s feet At t second, s feet At t seconds, s feet At t seconds, s feet In Eercises 8, find the equation of the parabola a b c that passes through the points To verif our result, use a graphing utilit to plot the points and graph the parabola,,,,,,,,,,,,,,,,,,,, 7,,,,, 8,,,,, In Eercises 9, find the equation of the circle D E F that passes through the points To verif our result, use a graphing utilit to plot the points and graph the circle 9,,,,,,,,,,,,,,, 8,,,,,

88 Chapter Sstems of Equations and Matrices Sports In Super Bowl I, on Januar, 97, the Green Ba Packers defeated the Kansas Cit Chiefs b a score of to The total points scored came from different scoring plas, which were a combination of touchdowns, etra-point kicks, and field goals, worth,, and points, respectivel The same number of touchdowns and etra-point kicks were scored There were si times as man touchdowns as field goals How man touchdowns, etra-point kicks, and field goals were scored during the game? (Source: Super Bowlcom) Sports In the 8 Women s NCAA Final Four Championship game, the Universit of Tennessee Lad Volunteers defeated the Universit of Stanford Cardinal b a score of to 8 The Lad Volunteers won b scoring a combination of two-point baskets, three-point baskets, and one-point free throws The number of two-point baskets was two more than the number of free throws The number of free throws was two more than five times the number of three-point baskets What combination of scoring accounted for the Lad Volunteers points? (Source: National Collegiate Athletic Association) Agriculture A miture of liters of chemical A, liters of chemical B, and liters of chemical C is required to kill a destructive crop insect Commercial spra X contains,, and parts, respectivel, of these chemicals Commercial spra Y contains onl chemical C Commercial spra Z contains onl chemicals A and B in equal amounts How much of each tpe of commercial spra is needed to get the desired miture? Acid Miture A chemist needs liters of a % acid solution The solution is to be mied from three solutions whose concentrations are %, %, and % How man liters of each solution will satisf each condition? (a) Use liters of the % solution (b) Use as little as possible of the % solution (c) Use as much as possible of the % solution 7 Finance A small corporation borrowed $77, to epand its clothing line Some of the mone was borrowed at 8%, some at 9%, and some at % How much was borrowed at each rate if the annual interest owed was $7, and the amount borrowed at 8% was four times the amount borrowed at %? 8 Finance A small corporation borrowed $8, to epand its line of tos Some of the mone was borrowed at 8%, some at 9%, and some at % How much was borrowed at each rate if the annual interest owed was $7, and the amount borrowed at 8% was five times the amount borrowed at %? Investment Portfolio In Eercises 9 and 7, consider an investor with a portfolio totaling $, that is invested in certificates of deposit, municipal bonds, blue-chip stocks, and growth or speculative stocks How much is invested in each tpe of investment? 9 The certificates of deposit pa % annuall, and the municipal bonds pa % annuall Over a five-ear period, the investor epects the blue-chip stocks to return 8% annuall and the growth stocks to return % annuall The investor wants a combined annual return of % and also wants to have onl one-fourth of the portfolio invested in stocks 7 The certificates of deposit pa % annuall, and the municipal bonds pa % annuall Over a five-ear period, the investor epects the blue-chip stocks to return % annuall and the growth stocks to return % annuall The investor wants a combined annual return of % and also wants to have onl one-fourth of the portfolio invested in stocks 7 Pulle Sstem A sstem of pulles is loaded with 8-pound and -pound weights (see figure) The tensions t and t in the ropes and the acceleration a of the -pound weight are found b solving the sstem of equations t t t where t and t are measured in pounds and a is measured in feet per second squared t lb t 8 lb (a) Solve this sstem (b) The -pound weight in the pulle sstem is replaced b a -pound weight The new pulle sstem will be modeled b the following sstem of equations t t t a a t t 8 a a 8 Solve this sstem and use our answer for the acceleration to describe what (if anthing) is happening in the pulle sstem

Multivariable Linear Sstems 88 7 Electrical Network Appling Kirchhoff s Laws to the electrical network in the figure, the currents I, I, and I are the solution of the sstem I I I I I 7 I I 8 Find the currents Ω I I Ω 7 Data Analsis: Stopping Distance In testing a new automobile braking sstem, the speed (in miles per hour) and the stopping distance (in feet) were recorded in the table Speed, Stopping Distance, 88 (a) Find a quadratic equation that models the data (b) Graph the model and the data on the same set of aes (c) Use the model to estimate the stopping distance when the speed is 7 miles per hour 7 Data Analsis: Wildlife A wildlife management team studied the reproduction rates of deer in three tracts of a wildlife preserve Each tract contained acres In each tract, the number of females, and the percent of females that had offspring the following ear, were recorded The results are shown in the table Number, Percent, 7 8 I (a) Find a quadratic equation that models the data (b) Use a graphing utilit to graph the model and the data in the same viewing window (c) Use the model to create a table of estimated values of Compare the estimated values with the actual data (d) Use the model to estimate the percent of females that had offspring when there were 7 females (e) Use the model to estimate the number of females when % of the females had offspring Ω 7 volts 8 volts Advanced Applications In Eercises 7 78, find values of,, and that satisf the sstem These sstems arise in certain optimization problems, and is called a Lagrange multiplier 7 7 77 78 True or False? In Eercises 79 8, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 79 The sstem z z z is in row-echelon form 8 If a sstem of three linear equations is inconsistent, then its graph has no points common to all three equations 8 The sstem z z z is consistent and has a unique solution In Eercises 8 8, find two sstems of linear equations that have the ordered triple as a solution (There are man correct answers) 8,, 8,, 8,, 7 8,, 7 CAPSTONE 8 Find values of a, b, and c (if possible) such that the sstem of linear equations has (a) a unique solution, (b) no solution, and (c) an infinite number of solutions a b z z cz

88 Chapter Sstems of Equations and Matrices Sstems of Inequalities Sketch the graphs of inequalities in two variables Solve sstems of inequalities Use sstems of inequalities in two variables to model and solve real-life problems The Graph of an Inequalit The statements < and are inequalities in two variables An ordered pair a, b is a solution of an inequalit in and if the inequalit is true when a and b are substituted for and, respectivel The graph of an inequalit is the collection of all solutions of the inequalit To sketch the graph of an inequalit, begin b sketching the graph of the corresponding equation The graph of the equation will normall separate the plane into two or more regions In each such region, one of the following must be true All points in the region are solutions of the inequalit No point in the region is a solution of the inequalit So, ou can determine whether the points in an entire region satisf the inequalit b simpl testing one point in the region GUIDELINES FOR SKETCHING THE GRAPH OF AN INEQUALITY IN TWO VARIABLES Replace the inequalit sign b an equal sign, and sketch the graph of the resulting equation (Use a dashed line for < or > and a solid line for or ) Test one point in each of the regions formed b the graph in Step If the point satisfies the inequalit, shade the entire region to denote that ever point in the region satisfies the inequalit EXAMPLE Sketching the Graph of an Inequalit Sketch the graph of Solution Begin b graphing the corresponding equation, which is a parabola, as shown in Figure 8 B testing a point above the parabola, and a point below the parabola,, ou can see that the points that satisf the inequalit are those ling above (or on) the parabola = STUDY TIP Note that when sketching the graph of an inequalit in two variables, a dashed line means all points on the line or curve are not solutions of the inequalit A solid line means all points on the line or curve are solutions of the inequalit Figure 8 (, ) Test point above parabola Test point below parabola (, )

Sstems of Inequalities 887 The inequalit in Eample is a nonlinear inequalit in two variables Most of the following eamples involve linear inequalities such as a b < c ( a and b are not both zero) The graph of a linear inequalit is a half-plane ling on one side of the line a b c EXAMPLE Sketching the Graph of a Linear Inequalit TECHNOLOGY A graphing utilit can be used to graph an inequalit or a sstem of inequalities For instance, to graph, enter and use the shade feature of the graphing utilit to shade the correct part of the graph You should obtain the graph in Figure Consult the user s guide for our graphing utilit for specific kestrokes Sketch the graph of each linear inequalit a b > Solution a The graph of the corresponding equation is a vertical line The points that satisf the inequalit > are those ling to the right of this line, as shown in Figure 9 b The graph of the corresponding equation is a horizontal line The points that satisf the inequalit are those ling below (or on) this line, as shown in Figure = > = Figure Figure 9 Figure < (, ) = Figure EXAMPLE Sketching the Graph of a Linear Inequalit Sketch the graph of < Solution The graph of the corresponding equation is a line, as shown in Figure Because the origin, satisfies the inequalit, the graph consists of the half-plane ling above the line (Check a point below the line Regardless of which point ou choose, ou will see that it does not satisf the inequalit) To graph a linear inequalit, it can help to write the inequalit in slope-intercept form For instance, b writing < in the form > ou can see that the solution points lie above the line or, as shown in Figure

888 Chapter Sstems of Equations and Matrices Sstems of Inequalities Man practical problems in business, science, and engineering involve sstems of linear inequalities A solution of a sstem of inequalities in and is a point, that satisfies each inequalit in the sstem To sketch the graph of a sstem of inequalities in two variables, first sketch the graph of each individual inequalit (on the same coordinate sstem) and then find the region that is common to ever graph in the sstem This region represents the solution set of the sstem For sstems of linear inequalities, it is helpful to find the vertices of the solution region EXAMPLE Solving a Sstem of Inequalities Sketch the graph (and label the vertices) of the solution set of the sstem Inequalit Inequalit < > Inequalit Solution The graphs of these inequalities are shown in Figures, 9, and, respectivel, on page 887 The triangular region common to all three graphs can be found b superimposing the graphs on the same coordinate sstem, as shown in Figure To find the vertices of the region, solve the three sstems of corresponding equations obtained b taking pairs of equations representing the boundaries of the individual regions Verte A:, Verte B:, Verte C:, = C = (, ) B = (, ) = = Solution set A = (, ) Figure Note in Figure that the vertices of the region are represented b open dots This means that the vertices are not solutions of the sstem of inequalities STUDY TIP Using different colored pencils to shade the solution of each inequalit in a sstem will make identifing the solution of the sstem of inequalities easier

Sstems of Inequalities 889 Not a verte For the triangular region shown in Figure, each point of intersection of a pair of boundar lines corresponds to a verte With more complicated regions, two border lines can sometimes intersect at a point that is not a verte of the region, as shown in Figure To keep track of which points of intersection are actuall vertices of the region, ou should sketch the region and refer to our sketch as ou find each point of intersection EXAMPLE Solving a Sstem of Inequalities Figure = = + (, ) Figure (, ) Sketch the region containing all points that satisf the sstem of inequalities Inequalit Inequalit Solution As shown in Figure, the points that satisf the inequalit Inequalit are the points ling above (or on) the parabola given b Parabola The points satisfing the inequalit Inequalit are the points ling below (or on) the line given b Line To find the points of intersection of the parabola and the line, solve the sstem of corresponding equations Using the method of substitution, ou can find the solutions to be, and, So, the region containing all points that satisf the sstem is indicated b the shaded region in Figure + > + < Figure When solving a sstem of inequalities, ou should be aware that the sstem might have no solution, as shown in Eample, or it might be represented b an unbounded region in the plane EXAMPLE A Sstem with No Solution Sketch the solution set of the sstem of inequalities Inequalit > < Inequalit Solution From the wa the sstem is written, it is clear that the sstem has no solution, because the quantit cannot be both less than and greater than Graphicall, the inequalit > is represented b the half-plane ling above the line, and the inequalit < is represented b the half-plane ling below the line, as shown in Figure These two half-planes have no points in common So, the sstem of inequalities has no solution

89 Chapter Sstems of Equations and Matrices Price p Consumer surplus Producer surplus Figure 7 Demand curve Suppl curve Number of units Equilibrium point Applications Eample in Section discussed the equilibrium point for a sstem of demand and suppl equations The net eample discusses two related concepts that economists call consumer surplus and producer surplus As shown in Figure 7, the consumer surplus is defined as the area of the region that lies below the demand curve, above the horizontal line passing through the equilibrium point, and to the right of the p-ais Similarl, the producer surplus is defined as the area of the region that lies above the suppl curve, below the horizontal line passing through the equilibrium point, and to the right of the p-ais The consumer surplus is a measure of the amount that consumers would have been willing to pa above what the actuall paid, whereas the producer surplus is a measure of the amount that producers would have been willing to receive below what the actuall received EXAMPLE 7 Consumer Surplus and Producer Surplus Price per unit (in dollars) 7 p Producer surplus Figure 8 Suppl vs Demand p = Consumer surplus p = + p =,,,, Number of units The demand and suppl equations for a new tpe of personal digital assistant are given b p Demand equation p Suppl equation where p is the price (in dollars) and represents the number of units Find the consumer surplus and producer surplus for these two equations Solution Begin b finding the equilibrium point (when suppl and demand are equal) b solving the equation In Eample in Section, ou saw that the solution is,, units, which corresponds to an equilibrium price of p $ So, the consumer surplus and producer surplus are the areas of the following triangular regions Consumer Surplus p p In Figure 8, ou can see that the consumer and producer surpluses are defined as the areas of the shaded triangles Consumer surplus (base)(height),, $,, Producer Surplus p p Producer surplus (base)(height),, $9,,

Sstems of Inequalities 89 EXAMPLE 8 Nutrition 8 (, ) (, ) (, ) (9, ) 8 Figure 9 The liquid portion of a diet is to provide at least calories, units of vitamin A, and 9 units of vitamin C A cup of dietar drink X provides calories, units of vitamin A, and units of vitamin C A cup of dietar drink Y provides calories, units of vitamin A, and units of vitamin C Set up a sstem of linear inequalities that describes how man cups of each drink should be consumed each da to meet or eceed the minimum dail requirements for calories and vitamins Solution Begin b letting and represent the following number of cups of dietar drink X number of cups of dietar drink Y To meet or eceed the minimum dail requirements, the following inequalities must be satisfied 9 Calories Vitamin A Vitamin C The last two inequalities are included because and cannot be negative The graph of this sstem of inequalities is shown in Figure 9 Eercises See wwwcalcchatcom for worked-out solutions to odd-numbered eercises In Eercises, fill in the blanks An ordered pair a, b is a of an inequalit in and if the inequalit is true when a and b are substituted for and, respectivel The of an inequalit is the collection of all solutions of the inequalit The graph of a inequalit is a half-plane ling on one side of the line a b c A of a sstem of inequalities in and is a point, that satisfies each inequalit in the sstem A of a sstem of inequalities in two variables is the region common to the graphs of ever inequalit in the sstem The area of the region that lies below the demand curve, above the horizontal line passing through the equilibrium point, to the right of the p-ais is called the In Eercises 7, sketch the graph of the inequalit 7 < 8 < 9 < > 7 < > 7 8 < 9 > 9 9 > In Eercises, use a graphing utilit to graph the inequalit < ln ln < 7 9 7 < 8 8 7 9 > 8 <

89 Chapter Sstems of Equations and Matrices In Eercises, write an inequalit for the shaded region shown in the figure 9 > > In Eercises, sketch the graph and label the vertices of the solution set of the sstem of inequalities < > > 7 7 > > < > > In Eercises 7, determine whether each ordered pair is a solution of the sstem of linear inequalities 7 > (a), (b), 8 (c), (d), 8 < (a), (b), < 7 (c) 8, (a), (c), 9 (a), 7 (c), (d), (b), (d), (b), (d), 8 7 8 < > < 9 > < 9 < In Eercises, use a graphing utilit to graph the solution set of the sstem of inequalities 7 8 < > 9 < In Eercises 7, derive a set of inequalities to describe the region 8 8 < > 9 > > < < < > e

Sstems of Inequalities 89 7 8 9 Rectangle: vertices at 7 Parallelogram: vertices at 7 Triangle: vertices at 7 Triangle: vertices at,, 9,, 9, 9,, 9,,,,,,,,,,,,,,,,, ( 8, 8 ) WRITING ABOUT CONCEPTS In Eercises 7 7, match the sstem of inequalities with the graph of its solution [The graphs are labeled (a), (b), (c), and (d)] (a) (b) (c) (d) 7 7 7 7 77 The graph of the solution of the inequalit < is shown in the figure Describe how the solution set would change for each of the following (a) (b) > CAPSTONE 78 (a) Eplain the difference between the graphs of the inequalit on the real number line and on the rectangular coordinate sstem (b) After graphing the boundar of the inequalit <, eplain how ou decide on which side of the boundar the solution set of the inequalit lies Suppl and Demand In Eercises 79 8, (a) graph the sstems representing the consumer surplus and producer surplus for the suppl and demand equations and (b) find the consumer surplus and producer surplus 79 8 8 8 Demand p p p p Suppl p p p 8 p 8 Production A furniture compan can sell all the tables and chairs it produces Each table requires hour in the assembl center and hours in the finishing center Each chair requires hours in the assembl center and hours in the finishing center The compan s assembl center is available hours per da, and its finishing center is available hours per da Find and graph a sstem of inequalities describing all possible production levels 8 Inventor A store sells two models of laptop computers Because of the demand, the store stocks at least twice as man units of model A as of model B The costs to the store for the two models are $8 and $, respectivel The management does not want more than $, in computer inventor at an one time, and it wants at least four model A laptop computers and two model B laptop computers in inventor at all times Find and graph a sstem of inequalities describing all possible inventor levels 8 Investment Analsis A person plans to invest up to $, in two different interest-bearing accounts Each account is to contain at least $ Moreover, the amount in one account should be at least twice the amount in the other account Find and graph a sstem of inequalities to describe the various amounts that can be deposited in each account 8 Ticket Sales For a concert event, there are $ reserved seat tickets and $ general admission tickets There are reserved seats available, and fire regulations limit the number of paid ticket holders to The promoter must take in at least $7, in ticket sales Find and graph a sstem of inequalities describing the different numbers of tickets that can be sold

89 Chapter Sstems of Equations and Matrices 87 Shipping A warehouse supervisor is told to ship at least packages of gravel that weigh pounds each and at least bags of stone that weigh 7 pounds each The maimum weight capacit of the truck to be used is 7 pounds Find and graph a sstem of inequalities describing the numbers of bags of stone and gravel that can be shipped 88 Nutrition A dietitian is asked to design a special dietar supplement using two different foods Each ounce of food X contains units of calcium, units of iron, and units of vitamin B Each ounce of food Y contains units of calcium, units of iron, and units of vitamin B The minimum dail requirements of the diet are units of calcium, units of iron, and units of vitamin B (a) Write a sstem of inequalities describing the different amounts of food X and food Y that can be used (b) Sketch a graph of the region corresponding to the sstem in part (a) (c) Find two solutions of the sstem and interpret their meanings in the contet of the problem 89 Data Analsis: Merchandise The table shows the retail sales (in millions of dollars) for Aeropostale, Inc from through 7 (Source: Aeropostale, Inc) Year Retail Sales, 8 9 79 Year 7 Retail Sales, 9 99 (a) Use the regression feature of a graphing utilit to find a linear model for the data Let t represent the ear, with t corresponding to (b) The total retail sales for Aeropostale during this eight-ear period can be approimated b finding the area of the trapezoid bounded b the linear model ou found in part (a) and the lines, t, and t 7 Use a graphing utilit to graph this region (c) Use the formula for the area of a trapezoid to approimate the total retail sales for Aeropostale 9 Phsical Fitness Facilit An indoor running track is to be constructed with a space for eercise equipment inside the track (see figure) The track must be at least meters long, and the eercise space must have an area of at least square meters Figure for 9 (a) Find a sstem of inequalities describing the requirements of the facilit (b) Graph the sstem from part (a) True or False? In Eercises 9 and 9, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 9 The area of the figure defined b the sstem is 99 square units 9 The graph below shows the solution of the sstem 9 > Eercise equipment SECTION PROJECT 8 8 Area Bounded b Concentric Circles Two concentric circles have radii of and, where > (see figure) The area between the boundaries of the circles must be at least square units (a) Find a sstem of inequalities that describes the constraints on the circles (b) Use a graphing utilit to graph the sstem of inequalities in part (a) Graph the line in the same viewing window (c) Identif the graph of the line in relation to the boundar of the inequalit Eplain its meaning in the contet of the problem

Matrices and Sstems of Equations 89 Matrices and Sstems of Equations Write matrices and identif their orders Perform elementar row operations on matrices Use matrices to solve sstems of linear equations Matrices In this section, ou will stud a streamlined technique for solving sstems of linear equations This technique involves the use of a rectangular arra of real numbers called a matri (The plural of matri is matrices) DEFINITION OF A MATRIX If m and n are positive integers, an m n (read m b n ) matri is a rectangular arra Row Row Row Row m Column Column Column Column n a a a a n a a a a n a a a a ṇ a m a m a m a mn in which each entr, a ij, of the matri is a number An m n matri has m rows and n columns Matrices are usuall denoted b capital letters The entr in the ith row and jth column is denoted b the double subscript notation a ij For instance, a refers to the entr in the second row, third column A matri having m rows and n columns is said to be of order m n If m n, the matri is square of order m m or n n For a square matri, the entries a, a, a, are the main diagonal entries EXAMPLE Order of Matrices Determine the order of each matri a b c d 7 Solution a This matri has one row and one column The order of the matri is b This matri has one row and four columns The order of the matri is c This matri has two rows and two columns The order of the matri is d This matri has three rows and two columns The order of the matri is NOTE A matri that has onl one row is called a row matri, and a matri that has onl one column is called a column matri

89 Chapter Sstems of Equations and Matrices A matri derived from a sstem of linear equations (each written in standard form with the constant term on the right) is the augmented matri of the sstem Moreover, the matri derived from the coefficients of the sstem (but not including the constant terms) is the coefficient matri of the sstem Sstem: Augmented Matri: Coefficient Matri: z z z Note the use of for the missing coefficient of the -variable in the third equation, and also note the fourth column of constant terms in the augmented matri When forming either the coefficient matri or the augmented matri of a sstem, ou should begin b verticall aligning the variables in the equations and using zeros for the coefficients of the missing variables NOTE The vertical dots in an augmented matri separate the coefficients of the linear sstem from the constant terms EXAMPLE Writing an Augmented Matri Write the augmented matri for the sstem of linear equations w 9 z w z z w What is the order of the augmented matri? Solution Begin b rewriting the linear sstem and aligning the variables w 9 z w z w z Net, use the coefficients and constant terms as the matri entries Include zeros for the coefficients of the missing variables R 9 R R R The augmented matri has four rows and five columns, so it is a matri The notation R n is used to designate each row in the matri For eample, Row is represented b R

Matrices and Sstems of Equations 897 Elementar Row Operations In Section, ou studied three operations that can be used on a sstem of linear equations to produce an equivalent sstem Interchange two equations Multipl an equation b a nonzero constant Add a multiple of an equation to another equation In matri terminolog, these three operations correspond to elementar row operations An elementar row operation on an augmented matri of a given sstem of linear equations produces a new augmented matri corresponding to a new (but equivalent) sstem of linear equations Two matrices are row-equivalent if one can be obtained from the other b a sequence of elementar row operations ELEMENTARY ROW OPERATIONS Interchange two rows Multipl a row b a nonzero constant Add a multiple of a row to another row Although elementar row operations are simple to perform, the involve a lot of arithmetic Because it is eas to make a mistake, ou should get in the habit of noting the elementar row operations performed in each step so that ou can go back and check our work Notice how this is done in the following eamples EXAMPLE Elementar Row Operations TECHNOLOGY Most graphing utilities can perform elementar row operations on matrices After performing a row operation, the new row-equivalent matri that is displaed on our graphing utilit is stored in the answer variable You should use the answer variable and not the original matri for subsequent row operations a Interchange the first and second rows of the original matri b Multipl the first row of the original matri b Original Matri Original Matri New Row-Equivalent Matri R R R New Row-Equivalent Matri c Add times the first row of the original matri to the third row Original Matri New Row-Equivalent Matri R R 8 Note that the elementar row operation is written beside the row that is changed

898 Chapter Sstems of Equations and Matrices In Eample in Section, ou used Gaussian elimination with back-substitution to solve a sstem of linear equations The net eample demonstrates the matri version of Gaussian elimination The two methods are essentiall the same The basic difference is that with matrices ou do not need to keep writing the variables EXAMPLE Comparing Linear Sstems and Matri Operations STUDY TIP Arithmetic errors are often made when elementar row operations are performed Note the operation ou perform in each step so that ou can go back and check our work Linear Sstem Associated Augmented Matri 9 z 9 z 7 7 Add the first equation to the Add the first row to the second equation second row R R R R 9 z 9 z z 7 7 Add times the first equation Add times the first row to the third equation to the third row R R R R 9 z 9 z z Add the second equation to the Add the second row to the third equation third row R R R R 9 z 9 z z Multipl the third equation b Multipl the third row b R 9 z 9 z R z At this point, ou can determine that z and use back-substitution to find and Substitute for z Solve for 9 Substitute for and for z Solve for The solution is,, and z NOTE Remember that ou should check a solution b substituting the values of,, and z into each equation of the original sstem For eample, ou can check the solution to Eample as follows Equation : 9 Equation : Equation : 7

Matrices and Sstems of Equations 899 The last matri in Eample is said to be in row-echelon form The term echelon refers to the stair-step pattern formed b the nonzero elements of the matri To be in this form, a matri must have the following properties ROW-ECHELON FORM AND REDUCED ROW-ECHELON FORM A matri in row-echelon form has the following properties An rows consisting entirel of zeros occur at the bottom of the matri For each row that does not consist entirel of zeros, the first nonzero entr is (called a leading ) For two successive (nonzero) rows, the leading in the higher row is farther to the left than the leading in the lower row A matri in row-echelon form is in reduced row-echelon form if ever column that has a leading has zeros in ever position above and below its leading It is worth noting that the row-echelon form of a matri is not unique That is, two different sequences of elementar row operations ma ield different row-echelon forms However, the reduced row-echelon form of a given matri is unique EXAMPLE Row-Echelon Form Determine whether each matri is in row-echelon form If it is, determine whether the matri is in reduced row-echelon form a b c d e f Solution The matrices in (a), (c), (d), and (f) are in row-echelon form The matrices in (d) and (f) are in reduced row-echelon form because ever column that has a leading has zeros in ever position above and below its leading The matri in (b) is not in row-echelon form because a row of all zeros does not occur at the bottom of the matri The matri in (e) is not in row-echelon form because the first nonzero entr in Row is not a leading Ever matri is row-equivalent to a matri in row-echelon form For instance, in Eample, ou can change the matri in part (e) to row-echelon form b multipling its second row b

9 Chapter Sstems of Equations and Matrices Gaussian Elimination with Back-Substitution Gaussian elimination with back-substitution works well for solving sstems of linear equations b hand or with a computer For this algorithm, the order in which the elementar row operations are performed is important You should operate from left to right b columns, using elementar row operations to obtain zeros in all entries directl below the leading s EXAMPLE Gaussian Elimination with Back-Substitution Solve the sstem Solution R R 7 9 R R R R R R 9 R R The matri is now in row-echelon form, and the corresponding sstem is z z z w w w z z z 7z 7 w w w 9 Using back-substitution, ou can determine that the solution is,, z, and w 9 Write augmented matri Interchange R and R so first column has leading in upper left corner Perform operations on R and R so first column has zeros below its leading Perform operations on R so second column has zeros below its leading Perform operations on R and R so third and fourth columns have leading s

Matrices and Sstems of Equations 9 You can use the following guidelines to solve a sstem of linear equations using Gaussian elimination with back-substitution GUIDELINES FOR SOLVING A SYSTEM OF LINEAR EQUATIONS USING GAUSSIAN ELIMINATION WITH BACK-SUBSTITUTION Write the augmented matri of the sstem of linear equations Use elementar row operations to rewrite the augmented matri in row-echelon form Write the sstem of linear equations corresponding to the matri in row-echelon form, and use back-substitution to find the solution When solving a sstem of linear equations, remember that it is possible for the sstem to have no solution If, in the elimination process, ou obtain a row of all zeros ecept for the last entr, it is unnecessar to continue the elimination process You can simpl conclude that the sstem has no solution, or is inconsistent EXAMPLE 7 A Sstem with No Solution Solve the sstem Solution R R R R R R R R Write augmented matri Perform row operations Perform row operations Note that the third row of this matri consists entirel of zeros ecept for the last entr This means that the original sstem of linear equations is inconsistent You can see wh this is true b converting back to a sstem of linear equations z z 7z z z z z 7 7 Because the third equation is not possible, the sstem has no solution

9 Chapter Sstems of Equations and Matrices Gauss-Jordan Elimination With Gaussian elimination, elementar row operations are applied to a matri to obtain a (row-equivalent) row-echelon form of the matri A second method of elimination, called Gauss-Jordan elimination, after Carl Friedrich Gauss and Wilhelm Jordan (8 899), continues the reduction process until a reduced row-echelon form is obtained This procedure is demonstrated in Eample 8 EXAMPLE 8 Gauss-Jordan Elimination STUDY TIP The advantage of using Gauss-Jordan elimination to solve a sstem of linear equations is that the solution of the sstem is easil found without using back-substitution, as illustrated in Eample 8 Use Gauss-Jordan elimination to solve the sstem z 9 z 7 Solution In Eample, Gaussian elimination was used to obtain the row-echelon form of the linear sstem above 9 Now, appl elementar row operations until ou obtain zeros above each of the leading s, as follows R R Perform operations on R so second column has a 9 9 zero above its leading 9R R Perform operations on R R R and R so third column has zeros above its leading The matri is now in reduced row-echelon form Converting back to a sstem of linear equations, ou have z Now ou can simpl read the solution,,, and z, which can be written as the ordered triple,, The elimination procedures described in this section sometimes result in fractional coefficients For instance, in the elimination procedure for the sstem z z 7 ou ma be inclined to multipl the first row b to produce a leading, which will result in working with fractional coefficients You can sometimes avoid fractions b judiciousl choosing the order in which ou appl elementar row operations

Matrices and Sstems of Equations 9 Recall from Section that when there are fewer equations than variables in a sstem of equations, then the sstem has either no solution or infinitel man solutions EXAMPLE 9 A Sstem with an Infinite Number of Solutions Solve the sstem z Solution The corresponding sstem of equations is Solving for and in terms of z, ou have and R R R R z z z z To write a solution of the sstem that does not use an of the three variables of the sstem, let a represent an real number and let z a Now substitute a for z in the equations for and z a z a R R So, the solution set can be written as an ordered triple with the form a, a, a where a is an real number Remember that a solution set of this form represents an infinite number of solutions Tr substituting values for a to obtain a few solutions Then check each solution in the original sstem of equations

9 Chapter Sstems of Equations and Matrices Eercises See wwwcalcchatcom for worked-out solutions to odd-numbered eercises In Eercises 8, fill in the blanks A rectangular arra of real numbers that can be used to solve a sstem of linear equations is called a A matri is if the number of rows equals the number of columns For a square matri, the entries a, a, a,, a nn are the entries A matri with onl one row is called a matri, and a matri with onl one column is called a matri The matri derived from a sstem of linear equations is called the matri of the sstem The matri derived from the coefficients of a sstem of linear equations is called the matri of the sstem 7 Two matrices are called if one of the matrices can be obtained from the other b a sequence of elementar row operations 8 A matri in row-echelon form is in if ever column that has a leading has zeros in ever position above and below its leading In Eercises 9, determine the order of the matri 9 7 8 7 7 7 7 9 In Eercises, write the augmented matri for the sstem of linear equations In Eercises, write the sstem of linear equations represented b the augmented matri (Use variables,, z, and w, if applicable) 7 7 8 7 9 7 8 8 z 8 7 z 8 z z 8z 9 9 z 7 z 9 8z z 9 In Eercises 7, fill in the blank(s) using elementar row operations to form a row-equivalent matri 7 8 9 7 7 8 8 7 8 8 8 7 7 8 9 7 8 8 8 8 8 7 9 8 7 9

Matrices and Sstems of Equations 9 In Eercises 8, identif the elementar row operation(s) being performed to obtain the new row-equivalent matri 7 8 Original Matri 8 7 9 Perform the sequence of row operations on the matri What did the operations accomplish? (a) Add times R to R (b) Add times R to R (c) Add times R to R (d) Multipl R b (e) Add times R to R Perform the sequence of row operations on the matri What did the operations accomplish? 7 (a) Add R to R (b) Interchange R and R (c) Add times R to R (d) Add 7 times R to R (e) Multipl R b (f) Add the appropriate multiples of to R, R, and R In Eercises, determine whether the matri is in row-echelon form If it is, determine if it is also in reduced row-echelon form 7 7 7 New Row-Equivalent Matri 7 9 R 9 8 7 7 7 8 8 7 In Eercises 8, write the matri in row-echelon form (Remember that the row-echelon form of a matri is not unique) 7 7 8 8 8 8 In Eercises 9, use the matri capabilities of a graphing utilit to write the matri in reduced row-echelon form 9 8 8 In Eercises 8, write the sstem of linear equations represented b the augmented matri Then use backsubstitution to solve (Use variables,, and z, if applicable) 7 8 9 9 8 In Eercises 9, an augmented matri that represents a sstem of linear equations (in variables,, and z, if applicable) has been reduced using Gauss-Jordan elimination Write the solution represented b the augmented matri 9 7 9 8 7

9 Chapter Sstems of Equations and Matrices In Eercises 8, use matrices to solve the sstem of equations (if possible) Use Gaussian elimination with back-substitution or Gauss-Jordan elimination 7 8 7 7 8 9 9 8 7 7 7 7 z z z 7 7 z z z 9 7 7 z 8 z z 77 78 79 8 z z z 8 8 z 8 7 z 8 8 8 In Eercises 8 9, use the matri capabilities of a graphing utilit to reduce the augmented matri corresponding to the sstem of equations, and solve the sstem 8 z 8 z z z z z 8z z 9 87 z w w z w z w 7 7 7 8 z z z z z z z z 7z 9 z 9 z z 9 8 8 88 z w z w z w z w 9 89 9 z w z w z In Eercises 9 9, determine whether the two sstems of linear equations ield the same solution If so, find the solution using matrices 9 (a) (b) z z z 9 (a) (b) z z z 9 (a) (b) z 7 7z z 8 9 (a) (b) z 9 z 8 z 98 Electrical Network The currents in an electrical network are given b the solution of the sstem I I I I I 8 I I z z z z w w z w z z z z z 8 z z z 8 WRITING ABOUT CONCEPTS 9 (a) Describe the row-echelon form of an augmented matri that corresponds to a sstem of linear equations that is inconsistent (b) Describe the row-echelon form of an augmented matri that corresponds to a sstem of linear equations that has an infinite number of solutions 9 Describe the three elementar row operations that can be performed on an augmented matri 97 What is the relationship between the three elementar row operations performed on an augmented matri and the operations that lead to equivalent sstems of equations? where I, I, and I are measured in amperes Solve the sstem of equations using matrices

Matrices and Sstems of Equations 97 99 Finance A small shoe corporation borrowed $,, to epand its line of shoes Some of the mone was borrowed at 7%, some at 8%, and some at % Use a sstem of equations to determine how much was borrowed at each rate if the annual interest was $, and the amount borrowed at % was times the amount borrowed at 7% Solve the sstem using matrices Finance A small software corporation borrowed $, to epand its software line Some of the mone was borrowed at 9%, some at %, and some at % Use a sstem of equations to determine how much was borrowed at each rate if the annual interest was $, and the amount borrowed at % was times the amount borrowed at 9% Solve the sstem using matrices In Eercises and, use a sstem of equations to find the equation of the parabola a b c that passes through the points Solve the sstem using matrices Use a graphing utilit to verif our results (, 9) (, ) 8 (, 8) (, ) (, ) (, 8) 8 8 8 8 Mathematical Modeling A video of the path of a ball thrown b a baseball plaer was analzed with a grid covering the TV screen The tape was paused three times, and the position of the ball was measured each time The coordinates obtained are shown in the table ( and are measured in feet) Horizontal Distance, Height, 9 (a) Use a sstem of equations to find the equation of the parabola a b c that passes through the three points Solve the sstem using matrices (b) Use a graphing utilit to graph the parabola (c) Graphicall approimate the maimum height of the ball and the point at which the ball struck the ground (d) Analticall find the maimum height of the ball and the point at which the ball struck the ground (e) Compare our results from parts (c) and (d) Data Analsis: Snowboarders The table shows the numbers of people (in millions) in the United States who participated in snowboarding in selected ears from to 7 (Source: National Sporting Goods Association) (a) Use a sstem of equations to find the equation of the parabola at bt c that passes through the points Let t represent the ear, with t corresponding to Solve the sstem using matrices (b) Use a graphing utilit to graph the parabola (c) Use the equation in part (a) to estimate the number of people who participated in snowboarding in 9 Does our answer seem reasonable? Eplain (d) Do ou believe that the equation can be used for ears far beond 7? Eplain True or False? In Eercises 7, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 7 is a matri The matri is in reduced row-echelon form 7 The method of Gaussian elimination reduces a matri until a reduced row-echelon form is obtained 8 Think About It The augmented matri below represents a sstem of linear equations (in variables,, and z) that has been reduced using Gauss-Jordan elimination Write a sstem of equations with nonzero coefficients that is represented b the reduced matri (There are man correct answers) Year 7 Number, CAPSTONE 9 In our own words, describe the difference between a matri in row-echelon form and a matri in reduced row-echelon form Include an eample of each to support our eplanation

98 Chapter Sstems of Equations and Matrices Operations with Matrices Determine whether two matrices are equal Add and subtract matrices and multipl matrices b scalars Multipl two matrices Use matri operations to model and solve real-life problems Equalit of Matrices In Section, ou used matrices to solve sstems of linear equations There is a rich mathematical theor of matrices, and its applications are numerous This section and the net two introduce some fundamentals of matri theor It is standard mathematical convention to represent matrices in an of the following three was REPRESENTATION OF MATRICES A matri can be denoted b an uppercase letter such as A, B, or C A matri can be denoted b a representative element enclosed in brackets, such as a ij, b ij, or c ij A matri can be denoted b a rectangular arra of numbers such as A a ij a a a a m a a a a m a a a a m a n a n a ṇ a mn Two matrices A a ij and B b ij are equal if the have the same order m n and a ij b ij for i m and j n In other words, two matrices are equal if their corresponding entries are equal EXAMPLE Equalit of Matrices Solve for a, a, a, and in the following matri equation a a a a a Solution Because two matrices are equal onl if their corresponding entries are equal, ou can conclude that a, a, a, and a Be sure ou see that for two matrices to be equal, the must have the same order and their corresponding entries must be equal For instance, but

Operations with Matrices 99 Matri Addition and Scalar Multiplication In this section, three basic matri operations will be covered The first two are matri addition and scalar multiplication With matri addition, ou can add two matrices (of the same order) b adding their corresponding entries DEFINITION OF MATRIX ADDITION If A a ij and B b ij are matrices of order m n, their sum is the m n matri given b A B a ij b ij The sum of two matrices of different orders is undefined EXAMPLE Addition of Matrices The Granger Collection ARTHUR CAYLEY (8 89) Cale, a Cambridge Universit graduate and a lawer b profession, invented matrices around 88 His groundbreaking work on matrices was begun as he studied the theor of transformations Cale also was instrumental in the development of determinants Cale and two American mathematicians, Benjamin Peirce (89 88) and his son Charles S Peirce (89 9), are credited with developing matri algebra a b c d The sum of A B and is undefined because A is of order and B is of order In operations with matrices, numbers are usuall referred to as scalars In this tet, scalars will alwas be real numbers You can multipl a matri A b a scalar c b multipling each entr in A b c DEFINITION OF SCALAR MULTIPLICATION If A a ij is an m n matri and c is a scalar, the scalar multiple of A b c is the m n matri given b ca ca ij

9 Chapter Sstems of Equations and Matrices EXPLORATION Consider matrices A, B, and C below Perform the indicated operations and compare the results A B 8 C 7,, a Find A B and B A b Find A B, then add C to the resulting matri Find B C, then add A to the resulting matri c Find A and B, then add the two resulting matrices Find A B, then multipl the resulting matri b The smbol A represents the negation of A, which is the scalar product A Moreover, if A and B are of the same order, then A B represents the sum of A and B That is, Subtraction of matrices The order of operations for matri epressions is similar to that for real numbers In particular, ou perform scalar multiplication before matri addition and subtraction, as shown in Eample (c) EXAMPLE Scalar Multiplication and Matri Subtraction For the following matrices, find (a) A, (b) B, and (c) A B A and B Solution A B A B a A Scalar multiplication Multipl each entr b 9 Simplif b B Definition of negation Multipl each entr b c A B 9 Matri subtraction 7 Subtract corresponding entries It is often convenient to rewrite the scalar multiple ca b factoring c out of ever entr in the matri For instance, in the following eample, the scalar has been factored out of the matri

Operations with Matrices 9 The properties of matri addition and scalar multiplication are similar to those of addition and multiplication of real numbers THEOREM PROPERTIES OF MATRIX ADDITION AND SCALAR MULTIPLICATION Let A, B, and C be m n matrices and let c and d be scalars A B B A Commutative Propert of Matri Addition A B C A B C Associative Propert of Matri Addition cda cda) Associative Propert of Scalar Multiplication A A Scalar Identit Propert ca B ca cb Distributive Propert c da ca da Distributive Propert Note that the Associative Propert of Matri Addition allows ou to write epressions such as A B C without ambiguit because the same sum occurs no matter how the matrices are grouped This same reasoning applies to sums of four or more matrices EXAMPLE Addition of More than Two Matrices B adding corresponding entries, ou obtain the following sum of four matrices EXAMPLE Using the Distributive Propert TECHNOLOGY Most graphing utilities have the capabilit of performing matri operations Consult the user s guide for our graphing utilit for specific kestrokes Tr using a graphing utilit to find the sum of the matrices and A B Perform the indicated matri operations Solution In Eample, ou could add the two matrices first and then multipl the matri b, as follows Notice that ou obtain the same result 7 7 7 7 9 8 7

9 Chapter Sstems of Equations and Matrices One important propert of addition of real numbers is that the number is the additive identit That is, c c for an real number c For matrices, a similar propert holds That is, if A is an m n matri and O is the m n zero matri consisting entirel of zeros, then A O A In other words, O is the additive identit for the set of all m n matrices For eample, the following matrices are the additive identities for the sets of all and matrices O and zero matri zero matri The algebra of real numbers and the algebra of matrices have man similarities For eample, compare the following solutions a b Real Numbers Solve for a a b a b a b a O m n Matrices Solve for X X A B X A A B A X O B A X B A STUDY TIP Remember that matrices are denoted b capital letters So, when ou solve for X, ou are solving for a matri that makes the matri equation true The algebra of real numbers and the algebra of matrices also have important differences, which will be discussed later EXAMPLE Solving a Matri Equation Solve for X in the equation X A B, where A and B Solution Begin b solving the matri equation for X to obtain X B A X B A Now, using the matrices A and B, ou have X Substitute the matrices Subtract matri A from matri B Multipl the matri b

Operations with Matrices 9 Matri Multiplication Another basic matri operation is matri multiplication At first glance, the definition ma seem unusual You will see later, however, that this definition of the product of two matrices has man practical applications DEFINITION OF MATRIX MULTIPLICATION If A a ij is an m n matri and B b ij is an n p matri, the product AB is an m p matri AB c ij where c ij a i b j a i b j a i b j a in b nj The definition of matri multiplication indicates a row-b-column multiplication, where the entr in the ith row and jth column of the product AB is obtained b multipling the entries in the ith row of A b the corresponding entries in the jth column of B and then adding the results So for the product of two matrices to be defined, the number of columns of the first matri must equal the number of rows of the second matri The general pattern for matri multiplication is as follows a a a a i a m a a a a i a m a a a a i a m a n a n a n a in a mn b b b b n b b b b n b j b j b j b nj b p b p b p b np c c c i c m c c c i c m c j c j c ij c mj c p c p c ip c mp a i b j a i b j a i b j a in b nj c ij EXAMPLE 7 Finding the Product of Two Matrices Find the product using A and AB B Solution To find the entries of the product, multipl each row of A b each column of B STUDY TIP In Eample 7, the product AB is defined because the number of columns of A is equal to the number of rows of B Also, note that the product AB has order AB 9

9 Chapter Sstems of Equations and Matrices Be sure ou understand that for the product of two matrices to be defined, the number of columns of the first matri must equal the number of rows of the second matri That is, the middle two indices must be the same The outside two indices give the order of the product, as shown below A m n Equal Order of AB B n p AB m p EXAMPLE 8 Finding the Product of Two Matrices EXPLORATION Use the following matrices to find AB, BA, ABC, and ABC What do our results tell ou about matri multiplication, commutativit, and associativit? A B C,, Find the product AB where A and Solution Note that the order of A is and the order of B is So, the product AB has order AB 7 B EXAMPLE 9 Patterns in Matri Multiplication a b 9 c The product AB for the following matrices is not defined because the number of columns of A is not equal to the number of rows of B and B A Not equal

Operations with Matrices 9 EXAMPLE Patterns in Matri Multiplication a b In Eample, note that the two products are different Even if both AB and BA are defined, matri multiplication is not, in general, commutative That is, for most matrices, AB BA This is one wa in which the algebra of real numbers and the algebra of matrices differ THEOREM PROPERTIES OF MATRIX MULTIPLICATION Let A, B, and C be matrices and let c be a scalar ABC ABC Associative Propert of Matri Multiplication AB C AB AC Distributive Propert A B)C AC BC Distributive Propert cab cab AcB Associative Propert of Scalar Multiplication DEFINITION OF IDENTITY MATRIX The n n matri that consists of s on its main diagonal and s elsewhere is called the identit matri of order n n and is denoted b I n Identit matri Note that an identit matri must be square When the order is understood to be n n, ou can denote simpl b I I n If A is an n n matri, the identit matri has the propert that AI n A and I n A A For eample, and AI A IA A

9 Chapter Sstems of Equations and Matrices Applications Matri multiplication can be used to represent a sstem of linear equations Note how the sstem a a a b a a a b a a a b STUDY TIP The column matri B is also called a constant matri Its entries are the constant terms in the sstem of equations can be written as the matri equation AX B, where A is the coefficient matri of the sstem, and X and B are column matrices b b a a a a a a a a a b A X B EXAMPLE Solving a Sstem of Linear Equations Consider the following sstem of linear equations STUDY TIP The notation A B represents the augmented matri formed when matri B is adjoined to matri A The notation I X represents the reduced row-echelon form of the augmented matri that ields the solution of the sstem a Write this sstem as a matri equation, AX B b Use Gauss-Jordan elimination on the augmented matri A B to solve for the matri X Solution a In matri form, AX B, the sstem can be written as follows b The augmented matri is formed b adjoining matri B to matri A A B Using Gauss-Jordan elimination, ou can rewrite this equation as I X So, the solution of the sstem of linear equations is,, and, and the solution of the matri equation is X

Operations with Matrices 97 EXAMPLE Softball Team Epenses STUDY TIP Notice in Eample that ou cannot find the total cost using the product EC because EC is not defined That is, the number of columns of E ( columns) does not equal the number of rows of C ( row) Two softball teams submit equipment lists to their sponsors Bats Balls 8 Gloves 7 Each bat costs $8, each ball costs $, and each glove costs $ Use matrices to find the total cost of equipment for each team Solution The equipment lists E and the costs per item C can be written in matri form as E Women s Team 8 and C 8 7 The total cost of equipment for each team is given b the product CE 8 8 8 8 7 8 8 7 Men s Team So, the total cost of equipment for the women s team is $ and the total cost of equipment for the men s team is $8 Eercises See wwwcalcchatcom for worked-out solutions to odd-numbered eercises In Eercises, fill in the blanks Two matrices are if all of their corresponding entries are equal When performing matri operations, real numbers are often referred to as A matri consisting entirel of zeros is called a matri and is denoted b The n n matri consisting of s on its main diagonal and s elsewhere is called the matri of order n n In Eercises and, match the matri propert with the correct form A, B, and C are matrices of order m n, and c and d are scalars (a) A A (b) A B C A B C (c) c da ca da (d) cda cda (e) A B B A (i) Distributive Propert (ii) Commutative Propert of Matri Addition (iii) Scalar Identit Propert (iv) Associative Propert of Matri Addition (v) Associative Propert of Scalar Multiplication (a) A O A (b) cab AcB (c) AB C AB AC (d) ABC ABC (i) Distributive Propert (ii) Additive Identit of Matri Addition (iii) Associative Propert of Matri Multiplication (iv) Associative Propert of Scalar Multiplication In Eercises 7, find and 7 8 7 7 8 8

98 Chapter Sstems of Equations and Matrices 9 In Eercises, if possible, find (a) A B, (b) A B, (c) A, and (d) A B 7 8 9 In Eercises, evaluate the epression 7 A A A A 8 A B A A A, 8,, 8, 8 7 8 9, B 8 9 7 9 8 B B B,,, B 7 7 B B 8 8 7 8 8 7 9 8 7 8 7 In Eercises 7, use the matri capabilities of a graphing utilit to evaluate the epression Round our results to three decimal places, if necessar 7 8 9 In Eercises, solve for X in the equation, given A [ and X A B X A B X A B A B X In Eercises, if possible, find AB and state the order of the result A 8 7 8 9 7 9 A 7 A A A A ] 8 9 89 9 889 978 8 7, B 8, B, B, B 7, B, B 8 7 9 B [ 9 9 7 8 ] 9 8 8

Operations with Matrices 99 In Eercises 8, use the matri capabilities of a graphing utilit to find AB, if possible 7 A 7, B 8 8 7 8 In Eercises 9, if possible, find (a) AB, (b) BA, and (c) A (Note: A AA ) 9 A, A A A A A 9 B A 9 A A 8, A, A, 7 A 8, B 8 8 8, B 8 7 7, 9 8, B 7 7 B, B B 8, 8 B 7 8 7, B A, B B 9, B B 8 8 In Eercises 8, evaluate the epression Use the matri capabilities of a graphing utilit to verif our answer 7 8 In Eercises 9, (a) write the sstem of linear equations as a matri equation, AX B, and (b) use Gauss-Jordan elimination on the augmented matri [A B] to solve for the matri X 9 9 9 7 7 8 9 7 7 8 WRITING ABOUT CONCEPTS In Eercises 7 7, let matrices A, B, C, and D be of orders,,, and, respectivel Determine whether the matrices are of proper order to perform the operation(s) If so, give the order of the answer 7 A C 8 B C 9 AB 7 BC 7 BC D 7 CB D 7 CAD 7 BCD

9 Chapter Sstems of Equations and Matrices WRITING ABOUT CONCEPTS (continued) 7 Let A and B be unequal diagonal matrices of the same order (A diagonal matri is a square matri in which each entr not on the main diagonal is zero) Determine the products AB for several pairs of such matrices Make a conjecture about a quick rule for such products 7 Eplain and correct the error in the matri addition a a a a a a a a a a a a a a a c c c a a a c 77 Manufacturing A corporation has three factories, each of which manufactures acoustic guitars and electric guitars The number of units of guitars produced at factor j in one da is represented b in the matri A 7 Find the production levels if production is increased b % 78 Manufacturing A corporation has four factories, each of which manufactures sport utilit vehicles and pickup trucks The number of units of vehicle i produced at factor j in one da is represented b in the matri A 9 a a a c 7 7 a a a c Find the production levels if production is increased b % 79 Agriculture A fruit grower raises two crops, apples and peaches Each of these crops is sent to three different outlets for sale These outlets are The Farmer s Market, The Fruit Stand, and The Fruit Farm The numbers of bushels of apples sent to the three outlets are,, and 7, respectivel The numbers of bushels of peaches sent to the three outlets are, 7, and, respectivel The profit per bushel for apples is $ and the profit per bushel for peaches is $ (a) Write a matri A that represents the number of bushels of each crop i that are shipped to each outlet j State what each entr a ij of the matri represents (b) Write a matri B that represents the profit per bushel of each fruit State what each entr b ij of the matri represents (c) Find the product BA and state what each entr of the matri represents a ij a ij 8 Revenue An electronics manufacturer produces three models of LCD televisions, which are shipped to two warehouses The numbers of units of model i that are shipped to warehouse j are represented b in the matri A,, 8, The prices per unit are represented b the matri B $999 $8999 $999 Compute BA and interpret the result 8 Inventor A compan sells five models of computers through three retail outlets The inventories are represented b S Model A B C D E S Outlet The wholesale and retail prices are represented b T Price Wholesale Retail $8 $ A $ $ B E T $ $ C Model $ $ D $ $ Compute ST and interpret the result 8 Labor/Wage Requirements A compan that manufactures boats has the following labor-hour and wage requirements Labor per boat Department Cutting Assembl Packaging Small Medium h h h S h h h h h h Large Wages per hour Plant A T $ $ $ B $ $ $,,, Cutting Assembl Packaging Department Compute ST and interpret the result a ij Boat size

Operations with Matrices 9 8 Profit At a local dair mart, the numbers of gallons of skim milk, % milk, and whole milk sold over the weekend are represented b A A 7 Skim % Whole milk milk milk The selling prices (in dollars per gallon) and the profits (in dollars per gallon) for the three tpes of milk sold b the dair mart are represented b B Selling price B $ $ $8 Profit (a) Compute AB and interpret the result (b) Find the dair mart s total profit from milk sales for the weekend 8 Profit At a convenience store, the numbers of gallons of 87-octane, 89-octane, and 9-octane gasoline sold over the weekend are represented b A Octane 87 89 9 Frida A Saturda 8 8 8 Sunda The selling prices (in dollars per gallon) and the profits (in dollars per gallon) for the three grades of gasoline sold b the convenience store are represented b B Selling price $ B $ $ Profit (a) Compute AB and interpret the result (b) Find the convenience store s profit from gasoline sales for the weekend 8 Eercise The numbers of calories burned b individuals of different bod weights performing different tpes of aerobic eercises for a -minute time period are shown in matri A -lb person A 9 7 8 9 $ $ $ $8 $9 $ Calories burned 7 8 Skim milk % milk Whole milk 87 89 Octane 9 -lb person Biccling 9 Jogging 79 Walking Frida Saturda Sunda (a) A -pound person and a -pound person biccled for minutes, jogged for minutes, and walked for minutes Organize the time the spent eercising in a matri B (b) Compute BA and interpret the result 8 Health Care The health care plans offered this ear b a local manufacturing plant are as follows For individuals, the comprehensive plan costs $9, the HMO standard plan costs $8, and the HMO Plus plan costs $898 For families, the comprehensive plan costs $7, the HMO standard plan costs $877, and the HMO Plus plan costs $8 The plant epects the costs of the plans to change net ear as follows For individuals, the costs for the comprehensive, HMO standard, and HMO Plus plans will be $89, $, and $997, respectivel For families, the costs for the comprehensive, HMO standard, and HMO Plus plans will be $998, $7, and $78, respectivel (a) Organize the information using two matrices A and B, where A represents the health care plan costs for this ear and B represents the health care plan costs for net ear State what each entr of each matri represents (b) Compute A B and interpret the result (c) The emploees receive monthl pachecks from which the health care plan costs are deducted Use the matrices from part (a) to write matrices that show how much will be deducted from each emploees pacheck this ear and net ear (d) Suppose instead that the costs of the health care plans increase b % net ear Write a matri that shows the new monthl paments True or False? In Eercises 87 and 88, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 87 Two matrices can be added onl if the have the same order 88 Matri multiplication is commutative 89 Find two matrices A and B such that AB BA CAPSTONE 9 Let matrices A and B be of orders and, respectivel Answer the following questions and eplain our reasoning (a) Is it possible that A B? (b) Is A B defined? (c) Is AB defined? If so, is it possible that AB BA?

9 Chapter Sstems of Equations and Matrices The Inverse of a Square Matri Verif that two matrices are inverses of each other Use Gauss-Jordan elimination to find the inverses of matrices Use a formula to find the inverses of matrices Use inverse matrices to solve sstems of linear equations The Inverse of a Matri This section further develops the algebra of matrices To begin, consider the real number equation a b To solve this equation for, multipl each side of the equation b a (provided that a ) a b a a a b a b a b The number a is called the multiplicative inverse of a because a a The definition of the multiplicative inverse of a matri is similar DEFINITION OF THE INVERSE OF A SQUARE MATRIX Let A be an n n matri and let I n be the n n identit matri If there eists a matri such that A AA I n A A then A is called the inverse of A The smbol A is read A inverse EXAMPLE The Inverse of a Matri Show that B is the inverse of A, where A and Solution To show that B is the inverse of A, show that AB I BA, as follows AB BA B As ou can see, AB I BA This is an eample of a square matri that has an inverse Note that not all square matrices have inverses NOTE Recall that it is not alwas true that AB BA, even if both products are defined However, if A and B are both square matrices and AB I n, it can be shown that BA I n So, in Eample, ou need onl to check that AB I

The Inverse of a Square Matri 9 Finding Inverse Matrices If a matri A has an inverse, A is called invertible (or nonsingular); otherwise, A is called singular A nonsquare matri cannot have an inverse To see this, note that if A is of order m n and B is of order n m (where m n), the products AB and BA are of different orders and so cannot be equal to each other Not all square matrices have inverses (see the matri at the bottom of page 9) If, however, a matri does have an inverse, that inverse is unique Eample shows how to use a sstem of equations to find the inverse of a matri EXAMPLE Finding the Inverse of a Matri Find the inverse of A Solution To find the inverse of A, tr to solve the matri equation AX I for X A Equating corresponding entries, ou obtain two sstems of linear equations X I Linear sstem with two variables, and Linear sstem with two variables, and Solve the first sstem using elementar row operations to determine that and From the second sstem ou can determine that and Therefore, the inverse of A is X A You can use matri multiplication to check this result Check AA A A

9 Chapter Sstems of Equations and Matrices TECHNOLOGY Most graphing utilities have the capabilit of finding the inverse of a matri Tr checking the result of Eample using a graphing utilit In Eample, note that the two sstems of linear equations have the same coefficient matri A Rather than solve the two sstems represented b and separatel, ou can solve them simultaneousl b adjoining the identit matri to the coefficient matri to obtain This doubl augmented matri can be represented as A I B appling Gauss-Jordan elimination to this matri, ou can solve both sstems with a single elimination process R R R R So, from the doubl augmented matri A I, ou obtain the matri I A A A I I This procedure (or algorithm) works for an square matri that has an inverse I A FINDING AN INVERSE MATRIX Let A be a square matri of order n Write the n n matri that consists of the given matri A on the left and the n n identit matri I on the right to obtain A I If possible, row reduce A to I using elementar row operations on the entire matri A I The result will be the matri I A If this is not possible, A is not invertible Check our work b multipling to see that AA I A A

The Inverse of a Square Matri 9 EXAMPLE Finding the Inverse of a Matri STUDY TIP Be sure to check our solution because it is eas to make algebraic errors when using elementar row operations Find the inverse of Solution Begin b adjoining the identit matri to A to form the matri A I Use elementar row operations to obtain the form I A, as follows R R R R R R R R R R R R So, the matri A is invertible and its inverse is A Confirm this result b multipling A and A to obtain I, as follows Check AA I A I A The process shown in Eample applies to an n n matri A When using this algorithm, if the matri A does not reduce to the identit matri, then A does not have an inverse For instance, the following matri has no inverse A To confirm that matri A above has no inverse, adjoin the identit matri to A to form A I and perform elementar row operations on the matri After doing so, ou will see that it is impossible to obtain the identit matri I on the left Therefore, A is not invertible

9 Chapter Sstems of Equations and Matrices The Inverse of a Matri Using Gauss-Jordan elimination to find the inverse of a matri works well (even as a computer technique) for matrices of order or greater For matrices, however, man people prefer to use a formula for the inverse rather than Gauss-Jordan elimination This simple formula, which works onl for matrices, is eplained as follows If A is a matri given b A a c then A is invertible if and onl if ad bc Moreover, if ad bc, the inverse is given b A b d ad bc d c b a Formula for inverse of matri A NOTE The denominator ad bc is called the determinant of the matri A You will stud determinants in the net section EXAMPLE Finding the Inverse of a Matri If possible, find the inverse of each matri a A b B Solution a For the matri A, appl the formula for the inverse of a matri to obtain ad bc Because this quantit is not zero, the inverse is formed b interchanging the entries on the main diagonal, changing the signs of the other two entries, and multipling b the scalar as follows A, b For the matri B, ou have ad bc Substitute for a, b, c, d, Multipl b the scalar and the determinant which means that B is not invertible EXPLORATION Use a graphing utilit with matri capabilities to find the inverse of the matri A What message appears on the screen? Wh does the graphing utilit displa this message?

The Inverse of a Square Matri 97 Sstems of Linear Equations You know that a sstem of linear equations can have eactl one solution, infinitel man solutions, or no solution If the coefficient matri A of a square sstem (a sstem that has the same number of equations as variables) is invertible, the sstem has a unique solution, as described in the following theorem THEOREM A SYSTEM OF EQUATIONS WITH A UNIQUE SOLUTION If A is an invertible matri, the sstem of linear equations represented b AX B has a unique solution given b X A B TECHNOLOGY To solve a sstem of equations with a graphing utilit, enter the matrices A and B in the matri editor Note that A must be a invertible matri Then, using the inverse ke, solve for X X A B The screen will displa the solution, matri X EXAMPLE Solving a Sstem Using an Inverse Matri You plan to invest $, in AAA-rated bonds, AA-rated bonds, and B-rated bonds and want an annual return of $7 The average ields are % on AAA bonds, 7% on AA bonds, and 9% on B bonds You will invest twice as much in AAA bonds as in B bonds Your investment can be represented as where,, and z represent the amounts invested in AAA, AA, and B bonds, respectivel Use an inverse matri to solve the sstem Solution Begin b writing the sstem in the matri form AX B 7 7 9 z 9z z Then, use Gauss-Jordan elimination to find A A 7, 7 z, 7 Finall, multipl B b A on the left to obtain the solution X A B 7, 7 The solution of the sstem is,, and z So, ou will invest $ in AAA bonds, $ in AA bonds, and $ in B bonds

98 Chapter Sstems of Equations and Matrices Eercises See wwwcalcchatcom for worked-out solutions to odd-numbered eercises In Eercises, fill in the blanks In a matri, the number of rows equals the number of columns If there eists an n n matri A such that AA I then A n A A, is called the of A If a matri A has an inverse, it is called invertible or ; if it does not have an inverse, it is called If A is an invertible matri, the sstem of linear equations represented b AX B has a unique solution given b X In Eercises, show that B is the inverse of A 7 8 9 A A A A A A A A In Eercises 8, find the inverse of the matri (if it eists) 7 8,,, 7 B, B B B B,, B, 7, B B 7 8 7 9 7 9 8 7 9 8 7 8 In Eercises 9, use the matri capabilities of a graphing utilit to find the inverse of the matri (if it eists) 9 7 7 7 7 9 8 7 7 8 7 9 In Eercises, use the formula on page 9 to find the inverse of the matri (if it eists) 7 7 9 7

The Inverse of a Square Matri 99 In Eercises 7, use the inverse matri found in Eercise to solve the sstem of linear equations 7 8 9 In Eercises and, use the inverse matri found in Eercise to solve the sstem of linear equations In Eercises and, use the inverse matri found in Eercise to solve the sstem of linear equations In Eercises and, use a graphing utilit to solve the sstem of linear equations using an inverse matri 7 z z z In Eercises 7, use an inverse matri to solve (if possible) the sstem of linear equations 7 8 8 9 8 9 8 9 z z z 88 8 7 z z z z z 8 z In Eercises 7 7, use the matri capabilities of a graphing utilit to solve (if possible) the sstem of linear equations 7 8 z z 7 8z 9 7 7 7 z z z 8 7 z z 9 z 7 WRITING ABOUT CONCEPTS 7 Write a brief paragraph eplaining the advantage of using inverse matrices to solve the sstems of linear equations in Eercises 7 7 In our own words, define the inverse of a square matri 7 What does it mean to sa that a matri is singular? 7 Consider matrices of the form A a w w z w w w z w z w w a 9 7 8 87 a 8 7 (a) Write a matri and a matri in the form of A Find the inverse of each (b) Use the result of part (a) to make a conjecture about the inverses of matrices in the form of A z 9z 7 9 7z a nn

9 Chapter Sstems of Equations and Matrices In Eercises 77 and 78, show that the matri is invertible and find its inverse 77 A sin cos 78 cos sin Investment Portfolio In Eercises 79 8, consider a person who invests in AAA-rated bonds, A-rated bonds, and B-rated bonds The average ields are % on AAA bonds, 7% on A bonds, and 9% on B bonds The person invests twice as much in B bonds as in A bonds Let,, and z represent the amounts invested in AAA, A, and B bonds, respectivel Use the inverse of the coefficient matri of this sstem to find the amount invested in each tpe of bond 79 $, $7 8 $, $7 8 $, $8 8 $, $8, Circuit Analsis In Eercises 8 and 8, consider the circuit in the figure The currents I, I, and I, in amperes, are the solution of the sstem of linear equations I I 7 Total Investment I I I I I where E and E are voltages Use the inverse of the coefficient matri of this sstem to find the unknown currents for the voltages I R Ω R Ω R + + _ E E _ I E E z 9z z total investment annual return Annual Return I A sec tan Ω tan sec 8 Enrollment The table shows the enrollment projections (in millions) for public universities in the United States for the ears through (Source: US National Center for Education Statistics, Digest of Education Statistics) Year Enrollment projections 89 (a) The data can be modeled b the quadratic function at bt c Create a sstem of linear equations for the data Let t represent the ear, with t corresponding to (b) Use the matri capabilities of a graphing utilit to find the inverse matri to solve the sstem from part (a) and find the least squares regression parabola at bt c (c) Use the graphing utilit to graph the parabola with the data (d) Do ou believe the model is a reasonable predictor of future enrollments? Eplain CAPSTONE 8 If A is a matri A a b then A is c d, invertible if and onl if ad bc If ad bc, verif that the inverse is A ad bc d c b a True or False? In Eercises 87 9, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 87 Multiplication of an invertible matri and its inverse is commutative 88 If ou multipl two square matrices and obtain the identit matri, ou can assume that the matrices are inverses of one another 89 All nonsquare matrices do not have inverses 9 The inverse of a nonsingular matri is unique 8 8 E volts, E 8 volts E volts, E volts 9 Writing Eplain how to determine whether the inverse of a matri eists If so, eplain how to find the inverse 9 Writing Eplain in our own words how to write a sstem of three linear equations in three variables as a matri equation, AX B, as well as how to solve the sstem using an inverse matri

7 The Determinant of a Square Matri 9 7 The Determinant of a Square Matri Find the determinants of matrices Find minors and cofactors of square matrices Find the determinants of square matrices Use the determinant to find the equation of a line through two points The Determinant of a Matri Ever square matri can be associated with a real number called its determinant Determinants have man uses, and several will be discussed in this section Historicall, the use of determinants arose from special number patterns that occur when sstems of linear equations are solved For instance, the sstem a b c a b c has a solution c b c b a b a b and provided that a b a b Note that the denominators of the two fractions are the same This denominator is called the determinant of the coefficient matri of the sstem Coefficient Matri A a a b b a c a c a b a b Determinant deta a b a b The determinant of the matri A can also be denoted b vertical bars on both sides of the matri, as indicated in the following definition DEFINITION OF THE DETERMINANT OF A MATRIX The determinant of the matri A a a b b is given b deta A a a b b a b a b A NOTE In this tet, deta and are used interchangeabl to represent the determinant of A Although vertical bars are also used to denote the absolute value of a real number, the contet will show which use is intended A convenient method for remembering the formula for the determinant of a matri is shown in the following diagram deta a a b b a b a b Note that the determinant is the difference of the products of the two diagonals of the matri

9 Chapter Sstems of Equations and Matrices EXAMPLE Determinant of a Matri EXPLORATION Use a graphing utilit with matri capabilities to find the determinant of the following matri A What message appears on the screen? Wh does the graphing utilit displa this message? Find the determinant of each matri a A b B c C Solution a b deta detb 7 c detc Notice in Eample that the determinant of a matri can be positive, zero, or negative The determinant of a matri of order is defined simpl as the entr of the matri For instance, if A, then deta TECHNOLOGY Most graphing utilities can evaluate the determinant of a matri For instance, ou can evaluate the determinant of a matri b entering the matri and then choosing the determinant feature Tr using a graphing utilit to check the determinants in Eample Minors and Cofactors To define the determinant of a square matri of order or higher, it is convenient to introduce the concepts of minors and cofactors MINORS AND COFACTORS OF A SQUARE MATRIX If A is a square matri, the minor M ij of the entr a ij is the determinant of the matri obtained b deleting the ith row and jth column of A The cofactor C ij of the entr is a ij C ij ij M ij NOTE In the sign pattern for cofactors, notice that odd positions (where i j is odd) have negative signs and even positions (where i j is even) have positive signs Sign Pattern for Cofactors matri matri n n matri

7 The Determinant of a Square Matri 9 EXAMPLE Finding the Minors and Cofactors of a Matri Find all the minors and cofactors of A Solution To find the minor M, delete the first row and first column of A and evaluate the determinant of the resulting matri Similarl, to find M, delete the first row and second column,, M M Continuing this pattern, ou obtain the minors M M M M M M 8 M M M Now, to find the cofactors, combine these minors with the checkerboard pattern of signs for a matri shown on page 9 C C C C C C 8 C C C The Determinant of a Square Matri The definition below is called inductive because it uses determinants of matrices of order n to define determinants of matrices of order n DETERMINANT OF A SQUARE MATRIX If A is a square matri (of order or greater), the determinant of A is the sum of the entries in an row (or column) of A multiplied b their respective cofactors For instance, epanding along the first row ields A a C a C a n C n Appling this definition to find a determinant is called epanding b cofactors NOTE This definition of the determinant ields for a matri A a a b b as previousl defined A a b a b

9 Chapter Sstems of Equations and Matrices EXAMPLE The Determinant of a Matri of Order Find the determinant of A Solution Note that this is the same matri as the one in Eample There ou found the cofactors of the entries in the first row to be C, C, and C So, b the definition of a determinant, ou have A a C a C a C First-row epansion When epanding b cofactors, ou do not need to find cofactors of zero entries, because zero times its cofactor is zero a ij C ij C ij So, the row (or column) containing the most zeros is usuall the best choice for epansion b cofactors This is demonstrated in the net eample EXAMPLE The Determinant of a Matri of Order Find the determinant of A Solution After inspecting this matri, ou can see that three of the entries in the third column are zeros So, ou can eliminate some of the work in the epansion b using the third column A C C C C Because C, C, and C have zero coefficients, ou need onl find the cofactor C To do this, delete the first row and third column of A and evaluate the determinant of the resulting matri C Epanding b cofactors in the second row ields C 8 7 So, ou obtain A C

7 The Determinant of a Square Matri 9 NOTE The method of finding the equation of a line works for all lines, including horizontal and vertical lines For instance, the equation of the vertical line through, and, is Application Given two points on a rectangular coordinate sstem, ou can find an equation of the line passing through the points using a determinant, as follows TWO-POINT FORM OF THE EQUATION OF A LINE An equation of the line passing through the distinct points and is given b,, (, ) (, ) Figure EXAMPLE Finding an Equation of a Line Find an equation of the line passing through the two points, and,, as shown in Figure Solution Let and Appling the determinant formula for the equation of a line produces,,,, To evaluate this determinant, ou can epand b cofactors along the first row to obtain the equation of the line as follows 7 Eercises See wwwcalcchatcom for worked-out solutions to odd-numbered eercises In Eercises, fill in the blanks A Both deta and represent the of the matri A The M ij of the entr a ij is the determinant of the matri obtained b deleting the ith row and jth column of the square matri A The C ij of the entr of the square matri A is given b ij M ij The method of finding the determinant of a matri of order or greater is called b In Eercises, find the determinant of the matri a ij 7 8 9 8 9 7 7 7 8 9 8 9 7

9 Chapter Sstems of Equations and Matrices In Eercises, use the matri capabilities of a graphing utilit to find the determinant of the matri 9 7 In Eercises, find all (a) minors and (b) cofactors of the matri 7 8 9 7 8 In Eercises, find the determinant of the matri b the method of epansion b cofactors Epand using the indicated row or column 7 8 (a) Row (a) Row (b) Column (b) Column (a) Row (a) Row (b) Column (b) Column 8 8 7 8 (a) Row (a) Row (b) Column (b) Column In Eercises 7 8, find the determinant of the matri Epand b cofactors on the row or column that appears to make the computations easiest 7 8 7 7 7 9 7 7 7 7 9 7 8 7 7 7 7 8 In Eercises 9, use the matri capabilities of a graphing utilit to evaluate the determinant 9, 8 7 8 9 7 8 8 7 8 8 9 7 8 8 In Eercises 8, find (a) A (b) B, (c) AB, and (d) AB A A A,, B B, B B A,

7 The Determinant of a Square Matri 97 7 A 8 A B, In Eercises 9, evaluate the determinant(s) to verif the equation 9 w z w c cz c w z z w a a b a z z a b a a a b a b a a b In Eercises, solve for 7 In Eercises 9 7, use a determinant to find an equation of the line passing through the points 9,,, 7,,, 7 7 8,, 7,,, 7,, B z z True or False? In Eercises 7 and 7, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 7 If a square matri has an entire row of zeros, the determinant will alwas be zero 7 If two columns of a square matri are the same, the determinant of the matri will be zero 7 Think About It If A is a matri of order such that is it possible to find Eplain A, A? WRITING ABOUT CONCEPTS 7 Write a brief paragraph eplaining the difference between a square matri and its determinant 8 Write a brief description eplaining the procedure for finding (a) the minor M ij and (b) the cofactor C ij of a square matri CAPSTONE 7 If A is an n n matri, eplain how to find the determinant of A SECTION PROJECT Cramer s Rule So far, ou have studied three methods for solving a sstem of linear equations: substitution, elimination with equations, and elimination with matrices Another method is Cramer s Rule, named after Gabriel Cramer (7 7) Cramer s Rule generalizes easil to sstems of n equations in n variables The value of each variable is given as the quotient of two determinants The denominator is the determinant of the coefficient matri, and the numerator is the determinant of the matri formed b replacing the column corresponding to the variable (being solved for) with the column representing the constants For instance, the solution for in the sstem a a a b a a a b a a a b is given b A A Cramer s Rule states that if a sstem of n linear equations in n variables has a coefficient matri A with a nonzero determinant A, the solution of the sstem is n A n A A, A, a a a a a a a a a a a a A, b b b a a a A where the ith column of A i is the column of constants in the sstem of equations If the determinant of the coefficient matri is zero, the sstem has either no solution or infinitel man solutions Use a graphing utilit and Cramer s Rule to solve (if possible) each sstem of equations (a) (b) z 7 z 8 z 9z 9 7z z 8 z (c) (d) 9z 7 z z z 9 7z

98 Chapter Sstems of Equations and Matrices CHAPTER SUMMARY Section Review Eercises Use the method of substitution to solve sstems of equations in two variables (p 8) Use the method of elimination to solve sstems of linear equations in two variables (p 8) 8 Interpret graphicall the numbers of solutions of sstems of linear equations in two variables 9 8 (p 87) Use sstems of linear equations in two variables to model and solve real-life problems (p 89) 9 Section Use back-substitution to solve linear sstems in row-echelon form (p 87), Use Gaussian elimination to solve sstems of linear equations (p 87) 7 Solve nonsquare sstems of linear equations (p 88), Use sstems of linear equations in three or more variables to model and solve real-life problems 8 (p 88) Section Sketch the graphs of inequalities in two variables (p 88) 9 Solve sstems of inequalities (p 888) Use sstems of inequalities in two variables to model and solve real-life problems (p 89) Section Write matrices and identif their orders (p 89) 7 Perform elementar row operations on matrices (p 897) 7, 7 Use matrices to solve sstems of linear equations (p 9) 7 9 Section Determine whether two matrices are equal (p 98) 9, 9 Add and subtract matrices and multipl matrices b scalars (p 99) 97 8 Multipl two matrices (p 9) 9 8 Use matri operations to model and solve real-life problems (p 9) 9, Section Verif that two matrices are inverses of each other (p 9) Use Gauss-Jordan elimination to find the inverses of matrices (p 9) 8 Use a formula to find the inverses of matrices (p 9) 9 Use inverse matrices to solve sstems of linear equations (p 97) 8 Section 7 Find the determinants of matrices (p 9) 9 Find minors and cofactors of square matrices (p 9) Find the determinants of square matrices (p 9) 7 Use the determinant to find the equation of a line through two points (p 9)

Review Eercises 99 REVIEW EXERCISES See wwwcalcchatcom for worked-out solutions to odd-numbered eercises In Eercises, solve the sstem b the method of substitution 8 7 7 7 8 9 9 In Eercises 8, solve the sstem b the method of elimination 8 9 7 8 8 In Eercises 9, use a graphing utilit to graph the lines in the sstem Use the graph to determine if the sstem is consistent or inconsistent If the sstem is consistent, determine the number of solutions 9 8 88 9 9 7 8 9 7 8 In Eercises, solve the sstem graphicall 8 8 In Eercises 7 and 8, use a graphing utilit to solve the sstem of equations Find the solution accurate to two decimal places 7 8 ln 9 7 8 8 9 Choice of Two Jobs You are offered two sales jobs at a pharmaceutical compan One compan offers an annual salar of $, plus a ear-end bonus of % of our total sales The other compan offers an annual salar of $, plus a ear-end bonus of % of our total sales What amount of sales will make the second offer better? Eplain Geometr The perimeter of a rectangle is inches The area of the rectangle is 9 square inches Find the dimensions of the rectangle Acid Miture Two hundred liters of a 7% acid solution is obtained b miing a 9% solution with a % solution How man liters of each must be used to obtain the desired miture? Fling Speeds Two airplanes leave Pittsburgh and Philadelphia at the same time, each going to the other cit One airplane flies miles per hour faster than the other Find the airspeed of each airplane if the cities are 7 miles apart and the airplanes pass one another after minutes of fling time Suppl and Demand In Eercises and, find the equilibrium point of the demand and suppl equations Demand p 7 p In Eercises and, use back-substitution to solve the sstem of linear equations z z z In Eercises 7, use Gaussian elimination to solve the sstem of equations Suppl p p 7 z 8 z z 9 z 7 z 7 8z 9z z In Eercises and, solve the nonsquare sstem of equations 7z 9z 7 z 9 8 z z z z 8 z 9 z 7z

9 Chapter Sstems of Equations and Matrices In Eercises and, find the equation of the parabola a b c that passes through the points To verif our result, use a graphing utilit to plot the points and graph the parabola In Eercises and, find the equation of the circle that passes through the points To verif our result, use a graphing utilit to plot the points and graph the circle (, ) (, ) (, ) (, ) D E F (, ) (, ) 7 Agriculture A miture of gallons of chemical A, 8 gallons of chemical B, and gallons of chemical C is required to kill a destructive crop insect Commercial spra X contains,, and parts, respectivel, of these chemicals Commercial spra Y contains onl chemical C Commercial spra Z contains chemicals A, B, and C in equal amounts How much of each tpe of commercial spra is needed to get the desired miture? 8 Vertical Motion An object moving verticall is at the given heights at the specified times Find the position function st gt v t s for the object (a) At t second, s feet At t seconds, s 8 feet At t seconds, s feet (b) At t second, s 8 feet At t seconds, s feet At t seconds, s feet In Eercises 9, sketch the graph of the inequalit 9 7 > (, ) (, ) 8 (, ) (, ) (, ) (, ) In Eercises, sketch the graph and label the vertices of the solution set of the sstem of inequalities 8 7 8 < > 9 8 In Eercises and, find a sstem of inequalities that models the description Use a graphing utilit to graph the solution set of the sstem Fruit Distribution A Pennslvania fruit grower has bushels of apples that are to be divided between markets in Harrisburg and Philadelphia These two markets need at least bushels and bushels, respectivel Inventor Costs A warehouse operator has, square feet of floor space in which to store two products Each unit of product I requires square feet of floor space and costs $ per da to store Each unit of product II requires square feet of floor space and costs $8 per da to store The total storage cost per da cannot eceed $, Suppl and Demand In Eercises and, (a) graph the sstems representing the consumer surplus and producer surplus for the suppl and demand equations and (b) find the consumer surplus and producer surplus p p In Eercises 8, determine the order of the matri Demand 7 7 8 8 8 9 9 Suppl p 7 p

Review Eercises 9 In Eercises 9 and 7, write the augmented matri for the sstem of linear equations 9 7 8 7 z z z In Eercises 7 and 7, write the sstem of linear equations represented b the augmented matri (Use variables,, z, and w, if applicable) 7 7 In Eercises 7 and 7, write the matri in row-echelon form (Remember that the row-echelon form of a matri is not unique) 8 7 7 In Eercises 7 78, write the sstem of linear equations represented b the augmented matri Then use back-substitution to solve the sstem (Use variables,, and z ) 7 7 77 78 9 In Eercises 79 88, use matrices and Gaussian elimination with back-substitution to solve the sstem of equations (if possible) 8 7 7 8 9 9 7 9 79 8 8 8 7 7 8 8 z z 8 8 z z z 87 88 In Eercises 89 9, use matrices and Gauss-Jordan elimination to solve the sstem of equations 89 z 9 z z 8z z 8z 9 9 In Eercises 9 and 9, use the matri capabilities of a graphing utilit to reduce the augmented matri corresponding to the sstem of equations, and solve the sstem 9 9 In Eercises 9 and 9, find and 9 9z z z 7z z z z w z w z w z 8w 8 z z z 8 z 9 9 7 z z z z z z z w w w 8 7 8 w w 9 9 z z z z z 9 z

9 Chapter Sstems of Equations and Matrices In Eercises 97 and 98, if possible, find (a) A B, (b) A B, (c) A, and (d) A B 97 98 In Eercises 99, perform the matri operations If it is not possible, eplain wh 99 In Eercises, solve for X in the equation, given and In Eercises 7 and 8, find AB, if possible 7 B A 7, 8 B A 7 In Eercises 9, perform the matri operations, if possible If it is not possible, eplain wh 9 A A 7 7 7 8 A [, ], 9 7 8 8, 8 B B 8 8 8 8 B [ 8 X A B X A B X A B A B X ] In Eercises 7 and 8, use the matri capabilities of a graphing utilit to find the product 7 8 9 Manufacturing A tire corporation has three factories, each of which manufactures two models of tires The number of units of model i produced at factor j in one da is represented b in the matri A 8 7 Find the production levels if production is decreased b % Cell Phone Charges The pa-as-ou-go charges (in dollars per minute) of two cellular telephone companies for calls inside the coverage area, regional roaming calls, and calls outside the coverage area are represented b C A C 7 8 Compan 8 B 9 Inside 8 Regional roaming Outside Coverage area Each month, ou plan to use minutes on calls inside the coverage area, 8 minutes on regional roaming calls, and minutes on calls outside the coverage area (a) Write a matri T that represents the times spent on the phone for each tpe of call (b) Compute TC and interpret the result a ij

Review Eercises 9 In Eercises, show that B is the inverse of A In Eercises 9, use the formula below to find the inverse of the matri, if it eists 9 7 8 In Eercises, use an inverse matri to solve (if possible) the sstem of linear equations 8 7 7 8 8 7 9 z z z 7 A 7 A A adbc[ d c,, b a] z 9 z z B 7 B 7 9 B A, A B 8, In Eercises 8, find the inverse of the matri (if it eists) 7 8 7 8 7 9 9 7 8 In Eercises 8, use the matri capabilities of a graphing utilit to solve (if possible) the sstem of linear equations 8 7 z z z 8 z z z z z 8 z z 8 7 z 9 z In Eercises 9, find the determinant of the matri 9 9 8 7 In Eercises, find all (a) minors and (b) cofactors of the matri 7 8 In Eercises 7, find the determinant of the matri Epand b cofactors on the row or column that appears to make the computations easiest 7 8 8 8 9 8 8 In Eercises, use a determinant to find an equation of the line passing through the points,,,,,,,,,,,, 9

9 Chapter Sstems of Equations and Matrices CHAPTER TEST Take this test as ou would take a test in class In Eercises, solve the sstem b the method of substitution 9 8 In Eercises, solve the linear sstem b the method of elimination z 7 7 7 8 z z In Eercises 7 9, sketch the graph and label the vertices of the solution of the sstem of inequalities 7 8 9 < > Write the augmented matri corresponding to the sstem of equations and solve the sstem z z z Find (a) A B, (b) A, (c) A B, and (d) AB (if possible) A, 8 B In Eercises and, find the inverse of the matri (if it eists) Use the result of Eercise to solve the sstem In Eercises 7, evaluate the determinant of the matri 7 8 7 8 A total of $, is invested in two funds paing % and % simple interest The earl interest is $9 How much is invested at each rate? 9 One hundred liters of a % solution is obtained b miing a % solution with a % solution How man liters of each solution must be used to obtain the desired miture?

PS Problem Solving 9 PS PROBLEM SOLVING Consider the sstem of equations b b (a) Use a graphing utilit to graph the sstem for b,,, and (b) For a fied even value of b >, make a conjecture about the number of points of intersection of the graphs in part (a) A theorem from geometr states that if a triangle is inscribed in a circle such that one side of the triangle is a diameter of the circle, then the triangle is a right triangle Show that this theorem is true for the circle and the triangle formed b the lines,, and Plot the points,,,,,, and, in a coordinate plane Draw the quadrilateral that has these four points as its vertices Write a sstem of linear inequalities that has the quadrilateral as its solution Eplain how ou found the sstem of inequalities Find square matrices A and B to demonstrate that A B A B The columns of matri T show the coordinates of the vertices of a triangle Matri A is a transformation matri A (a) Find AT and AAT Then sketch the original triangle and the two transformed triangles What transformation does A represent? (b) Given the triangle determined b AAT, describe the transformation process that produces the triangle determined b AT and then the triangle determined b T (a) The matri P From R D I 7 T 8 R D I is called a stochastic matri Each entr p ij i j represents the proportion of the voting population that changes from part i to part j, and p ii represents the proportion that remains loal to the part from one election to the net Compute and interpret P (b) Use a graphing utilit to find P, P, P, P, P 7, and P 8 the matri in part (a) Can ou detect a pattern? To 7 If a, b, and c are real numbers such that c and ac bc, then a b However, if A, B, and C are nonzero matrices such that AC BC, then A is not necessaril equal to B Illustrate this using the following matrices A 8 If a and b are real numbers such that ab, then a or b However, if A and B are matrices such that AB O, it is not necessaril true that A O or B O Illustrate this using the following matrices A 9 Let,, A B (a) Show that A A I, where I is the identit matri of order (b) Show that A I A (c) Show in general that for an square matri satisfing A A I the inverse of A is given b A I A B Let A and B Find AB, A B, and B A Make a conjecture about the inverses of two nonsingular matrices Check our conjecture using two different nonsingular matrices Let i and let A and B i i i i (a) Find A, A, and A Identif an similarities with i, i, and i (b) Find and identif B Use the sstem z z z 8, C to write two different matrices in row-echelon form that ield the same solution

9 Chapter Sstems of Equations and Matrices Consider square matrices in which the entries are consecutive integers An eample of such a matri is 7 A sstem of two equations in two unknowns is solved and has a finite number of solutions Determine the maimum number of solutions of the sstem satisfing each of the following (a) Both equations are linear (b) One equation is linear and the other is quadratic (c) Both equations are quadratic Three people were asked to solve a sstem of equations using an augmented matri Each person reduced the matri to row-echelon form The reduced matrices were and 8 9 (a) Use a graphing utilit to evaluate the determinants of four matrices of this tpe Make a conjecture based on the results (b) Verif our conjecture Find k and k such that the sstem of equations has an infinite number of solutions 8 k k,, Can all three be right? Eplain our reasoning In Eercises 7 and 8, use the following information The area of a triangle with vertices and is Area ±,,,,,, where the smbol ± indicates that the appropriate sign should be chosen to ield a positive area 7 Find a value of such that the triangle with vertices,,,, 8, has an area of 8 A large region of forest has been infested with gps moths The region is roughl triangular, as shown in the figure From the northernmost verte A of the region, the distances to the other vertices are miles south and miles east (for verte B), and miles south and 8 miles east (for verte C) Approimate the number of square miles in this region A mi mi mi B 8 mi 9 Find an eample of a singular matri satisfing A A Verif the following equation a b c a bb cc a a b c Verif the following equation a b c a bb cc aa b c a b c Verif the following equation a c b a b c Use the equation given in Eercise as a model to find a determinant that is equal to a b c d Three points and are collinear (lie on the same line) if and onl if,,,,, Find such that the points,,,, and, are collinear W C N S E