Matrices and Determinants

Transcription

1 Matrices and Determinants Matrices and Systems of Equations Operations with Matrices The Inverse of a Square Matrix The Determinant of a Square Matrix Applications of Matrices and Determinants Matrices can be used to analyze financial information such as the profit a fruit farmer makes on two fruit crops Darren McCollester/Getty Images SELECTED APPLICATIONS Matrices have many real-life applications The applications listed below represent a small sample of the applications in this chapter Electrical Network, Exercise, page Data Analysis: Snowboarders, Exercise 9, page Agriculture, Exercise, page 99 Profit, Exercise 7, page Investment Portfolio, Exercises 7 7, page 9 Data Analysis: Supreme Court, Exercise, page Long-Distance Plans, Exercise, page 7

2 7 Chapter Matrices and Determinants Matrices and Systems of Equations What you should learn Write matrices and identify their orders Perform elementary row operations on matrices Use matrices and Gaussian elimination to solve systems of linear equations Use matrices and Gauss- Jordan elimination to solve systems of linear equations Why you should learn it You can use matrices to solve systems of linear equations in two or more variables For instance, in Exercise 9 on page, you will use a matrix to find a model for the number of people who participated in snowboarding in the United States from 997 to Matrices In this section, you will study a streamlined technique for solving systems of linear equations This technique involves the use of a rectangular array of real numbers called a matrix The plural of matrix is matrices Definition of Matrix If m and n are positive integers, an m n (read m by n ) matrix is a rectangular array 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 entry, a ij, of the matrix is a number An m n matrix has m rows and n columns Matrices are usually denoted by capital letters The entry in the ith row and jth column is denoted by the double subscript notation a ij For instance, a refers to the entry in the second row, third column A matrix having m rows and n columns is said to be of order m n If m n, the matrix is square of order n For a square matrix, the entries a, a, a, are the main diagonal entries Example Order of Matrices Determine the order of each matrix a b c d 7 Solution a This matrix has one row and one column The order of the matrix is b This matrix has one row and four columns The order of the matrix is c This matrix has two rows and two columns The order of the matrix is d This matrix has three rows and two columns The order of the matrix is Now try Exercise A matrix that has only one row is called a row matrix, and a matrix that has only one column is called a column matrix

3 Section Matrices and Systems of Equations 7 The vertical dots in an augmented matrix separate the coefficients of the linear system from the constant terms A matrix derived from a system of linear equations (each written in standard form with the constant term on the right) is the augmented matrix of the system Moreover, the matrix derived from the coefficients of the system (but not including the constant terms) is the coefficient matrix of the system System: Augmented Matrix: Coefficient Matrix: x y x y x z z z Note the use of for the missing coefficient of the y-variable in the third equation, and also note the fourth column of constant terms in the augmented matrix When forming either the coefficient matrix or the augmented matrix of a system, you should begin by vertically aligning the variables in the equations and using zeros for the coefficients of the missing variables Example Writing an Augmented Matrix Write the augmented matrix for the system of linear equations x y w 9 y z w x z w x y z What is the order of the augmented matrix? Solution Begin by rewriting the linear system and aligning the variables x y w 9 y z w x z w x y z Next, use the coefficients and constant terms as the matrix entries Include zeros for the coefficients of the missing variables R 9 R R R The augmented matrix has four rows and five columns, so it is a matrix The notation R n is used to designate each row in the matrix For example, Row is represented by R Now try Exercise 9

4 7 Chapter Matrices and Determinants Elementary Row Operations In Section 7, you studied three operations that can be used on a system of linear equations to produce an equivalent system Interchange two equations Multiply an equation by a nonzero constant Add a multiple of an equation to another equation In matrix terminology, these three operations correspond to elementary row operations An elementary row operation on an augmented matrix of a given system of linear equations produces a new augmented matrix corresponding to a new (but equivalent) system of linear equations Two matrices are row-equivalent if one can be obtained from the other by a sequence of elementary row operations Elementary Row Operations Interchange two rows Multiply a row by a nonzero constant Add a multiple of a row to another row Although elementary row operations are simple to perform, they involve a lot of arithmetic Because it is easy to make a mistake, you should get in the habit of noting the elementary row operations performed in each step so that you can go back and check your work Example Elementary Row Operations Technology Most graphing utilities can perform elementary row operations on matrices Consult the user s guide for your graphing utility for specific keystrokes After performing a row operation, the new row-equivalent matrix that is displayed on your graphing utility is stored in the answer variable You should use the answer variable and not the original matrix for subsequent row operations a Interchange the first and second rows of the original matrix Original Matrix b Multiply the first row of the original matrix by Original Matrix c Add times the first row of the original matrix to the third row Original Matrix New Row-Equivalent Matrix R R Note that the elementary row operation is written beside the row that is changed Now try Exercise New Row-Equivalent Matrix R R New Row-Equivalent Matrix R

5 Section Matrices and Systems of Equations 7 In Example in Section 7, you used Gaussian elimination with backsubstitution to solve a system of linear equations The next example demonstrates the matrix version of Gaussian elimination The two methods are essentially the same The basic difference is that with matrices you do not need to keep writing the variables Example Comparing Linear Systems and Matrix Operations Remember that you should check a solution by substituting the values of x, y, and z into each equation of the original system For example, you can check the solution to Example as follows Equation : 9 Equation : Equation : 7 Linear System x y z 9 x y x y z 7 Add the first equation to the second equation x y z 9 y z x y z 7 Associated Augmented Matrix 9 7 Add the first row to the second row R R Add times the first equation Add times the first row to the third equation to the third row R R 9 x y z 9 y z y z Add the second equation to the third equation x y z 9 y z z Multiply the third equation by x y z 9 y z z At this point, you can use back-substitution to find x and y y y x 9 x Substitute for z Solve for y Multiply the third row by R Substitute for y and for z Solve for x The solution is x, y, and z Now try Exercise 7 R R R R R R R 9 7 Add the second row to the third row R R 9 9

6 7 Chapter Matrices and Determinants The last matrix in Example is said to be in row-echelon form The term echelon refers to the stair-step pattern formed by the nonzero elements of the matrix To be in this form, a matrix must have the following properties Row-Echelon Form and Reduced Row-Echelon Form A matrix in row-echelon form has the following properties Any rows consisting entirely of zeros occur at the bottom of the matrix For each row that does not consist entirely of zeros, the first nonzero entry 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 matrix in row-echelon form is in reduced row-echelon form if every column that has a leading has zeros in every position above and below its leading Example Row-Echelon Form Determine whether each matrix is in row-echelon form If it is, determine whether the matrix 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 every column that has a leading has zeros in every position above and below its leading The matrix in (b) is not in row-echelon form because a row of all zeros does not occur at the bottom of the matrix The matrix in (e) is not in row-echelon form because the first nonzero entry in Row is not a leading Now try Exercise 9 Every matrix is row-equivalent to a matrix in row-echelon form For instance, in Example, you can change the matrix in part (e) to row-echelon form by multiplying its second row by

7 Section Matrices and Systems of Equations 77 Gaussian Elimination with Back-Substitution Gaussian elimination with back-substitution works well for solving systems of linear equations by hand or with a computer For this algorithm, the order in which the elementary row operations are performed is important You should operate from left to right by columns, using elementary row operations to obtain zeros in all entries directly below the leading s Example Gaussian Elimination with Back-Substitution Solve the system Solution Write augmented matrix 7 9 R Interchange R and R so R first column has leading in upper left corner 7 9 Perform operations on R and R so first column has R R zeros below its leading R R Perform operations on R so second column has zeros below its leading R R 9 Perform operations on R and R so third and fourth columns have leading s R R The matrix is now in row-echelon form, and the corresponding system is x y z y z z x x x w w w y z y z y z y 7z w w w Using back-substitution, the solution is x, y, Now try Exercise 9 z, and w

8 7 Chapter Matrices and Determinants The procedure for using Gaussian elimination with back-substitution is summarized below Gaussian Elimination with Back-Substitution Write the augmented matrix of the system of linear equations Use elementary row operations to rewrite the augmented matrix in row-echelon form Write the system of linear equations corresponding to the matrix in row-echelon form, and use back-substitution to find the solution When solving a system of linear equations, remember that it is possible for the system to have no solution If, in the elimination process, you obtain a row with zeros except for the last entry, it is unnecessary to continue the elimination process You can simply conclude that the system has no solution, or is inconsistent Example 7 A System with No Solution Solve the system Solution x x x x R R R R R R R R Write augmented matrix Perform row operations Perform row operations Note that the third row of this matrix consists of zeros except for the last entry This means that the original system of linear equations is inconsistent You can see why this is true by converting back to a system of linear equations x y z y z y 7z y z z y z y z 7 7 Because the third equation is not possible, the system has no solution Now try Exercise 7

9 Gauss-Jordan Elimination Section Matrices and Systems of Equations 79 With Gaussian elimination, elementary row operations are applied to a matrix to obtain a (row-equivalent) row-echelon form of the matrix A second method of elimination, called Gauss-Jordan elimination, after Carl Friedrich Gauss and Wilhelm Jordan ( 99), continues the reduction process until a reduced row-echelon form is obtained This procedure is demonstrated in Example Example Gauss-Jordan Elimination Technology For a demonstration of a graphical approach to Gauss-Jordan elimination on a matrix, see the Visualizing Row Operations Program available for several models of graphing calculators at our website collegehmcocom The advantage of using Gauss- Jordan elimination to solve a system of linear equations is that the solution of the system is easily found without using back-substitution, as illustrated in Example Use Gauss-Jordan elimination to solve the system Solution In Example, Gaussian elimination was used to obtain the row-echelon form of the linear system above 9 Now, apply elementary row operations until you 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 and R so third column has R zeros above its leading The matrix is now in reduced row-echelon form Converting back to a system of linear equations, you have x y z Now you can simply read the solution, x, y, and z, which can be written as the ordered triple,, Now try Exercise 9 x y x y x y z z 9 7 The elimination procedures described in this section sometimes result in fractional coefficients For instance, in the elimination procedure for the system x y x y x y z z 7 you may be inclined to multiply the first row by to produce a leading, which will result in working with fractional coefficients You can sometimes avoid fractions by judiciously choosing the order in which you apply elementary row operations

10 Chapter Matrices and Determinants Recall from Chapter 7 that when there are fewer equations than variables in a system of equations, then the system has either no solution or infinitely many solutions Example 9 A System with an Infinite Number of Solutions In Example 9, x and y are solved for in terms of the third variable z To write a solution to the system that does not use any of the three variables of the system, let a represent any real number and let z a Then solve for x and y The solution can then be written in terms of a, which is not one of the variables of the system Solve the system x y z x y Solution R R R R R The corresponding system of equations is x z y z Solving for x and y in terms of z, you have x z and To write a solution to the system that does not use any of the three variables of the system, let a represent any real number and let z a R Now substitute a for z in the equations for x and y x z a y z a So, the solution set can be written as an ordered triple with the form a, a, a y z where a is any real number Remember that a solution set of this form represents an infinite number of solutions Try substituting values for a to obtain a few solutions Then check each solution in the original equation Now try Exercise It is worth noting that the row-echelon form of a matrix is not unique That is, two different sequences of elementary row operations may yield different row-echelon forms This is demonstrated in Example

11 Section Matrices and Systems of Equations Example Comparing Row-Echelon Forms Compare the following row-echelon form with the one found in Example Is it the same? Does it yield the same solution? R R R R R R Solution x y x y x y This row-echelon form is different from that obtained in Example The corresponding system of linear equations for this row-echelon matrix is x y y z z R R R R z z Using back-substitution on this system, you obtain the solution x, y, and z 9 7 which is the same solution that was obtained in Example Now try Exercise You have seen that the row-echelon form of a given matrix is not unique; however, the reduced row-echelon form of a given matrix is unique Try applying Gauss-Jordan elimination to the row-echelon matrix in Example to see that you obtain the same reduced row-echelon form as in Example

12 Chapter Matrices and Determinants Exercises VOCABULARY CHECK: Fill in the blanks A rectangular array of real numbers than can be used to solve a system of linear equations is called a A matrix is if the number of rows equals the number of columns For a square matrix, the entries a, a, a,, a nn are the entries A matrix with only one row is called a matrix and a matrix with only one column is called a matrix The matrix derived from a system of linear equations is called the matrix of the system The matrix derived from the coefficients of a system of linear equations is called the matrix of the system 7 Two matrices are called if one of the matrices can be obtained from the other by a sequence of elementary row operations A matrix in row-echelon form is in if every column that has a leading has zeros in every position above and below its leading 9 The process of using row operations to write a matrix in reduced row-echelon form is called In Exercises, determine the order of the matrix In Exercises 7, write the augmented matrix for the system of linear equations 7 7x y x y x y x 9y 9 x y z 7x z x y z x y z x y x y z 9x y z 7x y z 9x z y z In Exercises, write the system of linear equations represented by the augmented matrix (Use variables x, y, z, and w, if applicable) In Exercises 9, fill in the blank(s) using elementary row operations to form a row-equivalent matrix

13 Section Matrices and Systems of Equations In Exercises, identify the elementary row operation(s) being performed to obtain the new row-equivalent matrix Original Matrix Original Matrix 7 Perform the sequence of row operations on the matrix What did the operations accomplish? (a) Add times R to R (b) Add times R to R (c) Add times R to R (d) Multiply R by (e) Add times R to R Perform the sequence of row operations on the matrix What did the operations accomplish? (a) Add R to R (b) Interchange R and R (c) Add times R to R (d) Add 7 times R to R (e) Multiply R by (f) Add the appropriate multiples of to R, R, and R In Exercises 9, determine whether the matrix is in row-echelon form If it is, determine if it is also in reduced row-echelon form Original Matrix 7 Original Matrix 7 7 New Row-Equivalent Matrix 9 New Row-Equivalent Matrix New Row-Equivalent Matrix New Row-Equivalent Matrix 9 7 R In Exercises, write the matrix in row-echelon form (Remember that the row-echelon form of a matrix is not unique) 7 In Exercises 7, use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form In Exercises, write the system of linear equations represented by the augmented matrix Then use back-substitution to solve (Use variables x, y, and z, if applicable)

14 Chapter Matrices and Determinants In Exercises 7, an augmented matrix that represents a system of linear equations (in variables x, y, and z, if applicable) has been reduced using Gauss-Jordan elimination Write the solution represented by the augmented matrix 7 9 In Exercises 7, use matrices to solve the system of equations (if possible) Use Gaussian elimination with backsubstitution or Gauss-Jordan elimination x y 7 x y x y x y 7 x y 7 x y x y x y x y x y x y 9 x y 7 7 x y x y x y x y 9 z z x y y z x x y z x y z 7x y x y z x y z x y z x y z x y z 9 x y x y z x y z x y z y z x y z x y z x z x y 7z x y z x z x y z x 9y z 9 7 x y z w x 7y z 9w x y 7z w x 9y z w 9 7 x y x y x y x y x y x y In Exercises 7 7, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system 7 x y z x y z x y z x y z 7 x y z x y z x y z x y z x 7 x y z x y z x y z 7 x y x y x y x y z w w z w z w y z w x y z w x y z w x y z w w w x y z w x y w y z w In Exercises 77, determine whether the two systems of linear equations yield the same solution If so, find the solution using matrices 77 (a) (b) x y z y z x y z y z z z 7 (a) (b) x y y x y z y z z 79 (a) (b) x y z 7 y 7z z (a) (b) x y z 9 y z z Use the system x y z x y z x y z 9 x y z y z z x y z y z z to write two different matrices in row-echelon form that yield the same solution z z

15 Section Matrices and Systems of Equations Electrical Network The currents in an electrical network are given by the solution of the system I I I I I I where I, I, and I are measured in amperes Solve the system of equations using matrices Partial Fractions Use a system of equations to write the partial fraction decomposition of the rational expression Solve the system using matrices x x x Partial Fractions Use a system of equations to write the partial fraction decomposition of the rational expression Solve the system using matrices x x x Finance A small shoe corporation borrowed $,, to expand its line of shoes Some of the money was borrowed at 7%, some at %, and some at % Use a system 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 system using matrices Finance A small software corporation borrowed $, to expand its software line Some of the money was borrowed at 9%, some at %, and some at % Use a system 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 system using matrices In Exercises 7 and, use a system of equations to find the specified equation that passes through the points Solve the system using matrices Use a graphing utility to verify your results 7 Parabola: Parabola: y ax bx c I y (, ) (, ) (, ) A x A x x B x B x C x C x y ax bx c y (, 9) (, ) (, ) x 9 Mathematical Modeling A videotape of the path of a ball thrown by a baseball player was analyzed 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 ( x and y are measured in feet) Horizontal distance, x Height, y 9 (a) Use a system of equations to find the equation of the parabola y ax bx c that passes through the three points Solve the system using matrices (b) Use a graphing utility to graph the parabola (c) Graphically approximate the maximum height of the ball and the point at which the ball struck the ground (d) Analytically find the maximum height of the ball and the point at which the ball struck the ground (e) Compare your results from parts (c) and (d) Model It 9 Data Analysis: Snowboarders The table shows the numbers of people y (in millions) in the United States who participated in snowboarding for selected years from 997 to (Source: National Sporting Goods Association) Year Number, y (a) Use a system of equations to find the equation of the parabola y at bt c that passes through the points Let t represent the year, with t 7 corresponding to 997 Solve the system using matrices (b) Use a graphing utility to graph the parabola (c) Use the equation in part (a) to estimate the number of people who participated in snowboarding in How does this value compare with the actual value of million? (d) Use the equation in part (a) to estimate y in the year Is the estimate reasonable? Explain

16 Chapter Matrices and Determinants Network Analysis In Exercises 9 and 9, answer the questions about the specified network (In a network it is assumed that the total flow into each junction is equal to the total flow out of each junction) 9 Water flowing through a network of pipes (in thousands of cubic meters per hour) is shown in the figure (a) Solve this system using matrices for the water flow represented by x i, i,,, 7 (b) Find the network flow pattern when x and x 7 (c) Find the network flow pattern when x and x 9 The flow of traffic (in vehicles per hour) through a network of streets is shown in the figure (a) Solve this system using matrices for the traffic flow represented by x i, i,,, (b) Find the traffic flow when x and x (c) Find the traffic flow when x and x Synthesis x x x 7 True or False? In Exercises 9 9, determine whether the statement is true or false Justify your answer 9 7 is a matrix 9 The matrix is in reduced row-echelon form x x x x x x x x x 9 The method of Gaussian elimination reduces a matrix until a reduced row-echelon form is obtained 9 Think About It The augmented matrix represents a system of linear equations (in variables x, y, and z) that has been reduced using Gauss-Jordan elimination Write a system of equations with nonzero coefficients that is represented by the reduced matrix (There are many correct answers) 97 Think About It (a) Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that is inconsistent (b) Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that has an infinite number of solutions 9 Describe the three elementary row operations that can be performed on an augmented matrix 99 What is the relationship between the three elementary row operations performed on an augmented matrix and the operations that lead to equivalent systems of equations? Writing In your own words, describe the difference between a matrix in row-echelon form and a matrix in reduced row-echelon form Skills Review In Exercises, sketch the graph of the function Do not use a graphing utility fx x x x x fx x x x f x x gx x hx lnx f x ln x

17 Section Operations with Matrices 7 Operations with Matrices What you should learn Decide whether two matrices are equal Add and subtract matrices and multiply matrices by scalars Multiply two matrices Use matrix operations to model and solve real-life problems Why you should learn it Matrix operations can be used to model and solve real-life problems For instance, in Exercise 7 on page, matrix operations are used to analyze annual health care costs Equality of Matrices In Section, you used matrices to solve systems of linear equations There is a rich mathematical theory of matrices, and its applications are numerous This section and the next two introduce some fundamentals of matrix theory It is standard mathematical convention to represent matrices in any of the following three ways Representation of Matrices A matrix can be denoted by an uppercase letter such as A, B, or C A matrix can be denoted by a representative element enclosed in brackets, such as a ij, b ij, or c ij A matrix can be denoted by a rectangular array of numbers such as a a a a n a a a a n A a ij a a a a ṇ a mn a m a m a m Two matrices A a ij and B b ij are equal if they 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 Equality of Matrices Royalty-Free/Corbis Solve for a, a, a, and in the following matrix equation a a a a Solution Because two matrices are equal only if their corresponding entries are equal, you can conclude that a, a, a a, Now try Exercise and a Be sure you see that for two matrices to be equal, they must have the same order and their corresponding entries must be equal For instance, but

18 Chapter Matrices and Determinants Matrix Addition and Scalar Multiplication In this section, three basic matrix operations will be covered The first two are matrix addition and scalar multiplication With matrix addition, you can add two matrices (of the same order) by adding their corresponding entries The Granger Collection Historical Note Arthur Cayley ( 9), a British mathematician, invented matrices around Cayley was a Cambridge University graduate and a lawyer by profession His groundbreaking work on matrices was begun as he studied the theory of transformations Cayley also was instrumental in the development of determinants Cayley and two American mathematicians, Benjamin Peirce (9 ) and his son Charles S Peirce (9 9), are credited with developing matrix algebra a Definition of Matrix Addition If A a ij and B b ij are matrices of order m n, their sum is the m n matrix given by The sum of two matrices of different orders is undefined A B a ij b ij Example b c d The sum of Addition of Matrices A and B is undefined because A is of order and B is of order Now try Exercise 7(a) In operations with matrices, numbers are usually referred to as scalars In this text, scalars will always be real numbers You can multiply a matrix A by a scalar c by multiplying each entry in A by c Definition of Scalar Multiplication If A a ij is an m n matrix and c is a scalar, the scalar multiple of A by c is the m n matrix given by ca ca ij

19 Section Operations with Matrices 9 The symbol 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, A B A B Subtraction of matrices The order of operations for matrix expressions is similar to that for real numbers In particular, you perform scalar multiplication before matrix addition and subtraction, as shown in Example (c) Example Scalar Multiplication and Matrix Subtraction Exploration Consider matrices A, B, and C below Perform the indicated operations and compare the results A C 7, B, a Find A B and B A b Find A B, then add C to the resulting matrix Find B C, then add A to the resulting matrix c Find A and B, then add the two resulting matrices Find A B, then multiply the resulting matrix by For the following matrices, find (a) A, (b) B, and (c) A B A and B Solution A 9 a Scalar multiplication Now try Exercises 7(b), (c), and (d) Multiply each entry by Simplify b B Definition of negation Multiply each entry by c A B 9 7 Matrix subtraction Subtract corresponding entries It is often convenient to rewrite the scalar multiple ca by factoring c out of every entry in the matrix For instance, in the following example, the scalar has been factored out of the matrix

20 9 Chapter Matrices and Determinants The properties of matrix addition and scalar multiplication are similar to those of addition and multiplication of real numbers 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 A B C A B C Associative Property of Matrix Addition cda cda) A A Scalar Identity Property ca B ca cb Distributive Property c da ca da Distributive Property Commutative Property of Matrix Addition Associative Property of Scalar Multiplication Note that the Associative Property of Matrix Addition allows you to write expressions such as A B C without ambiguity 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 By adding corresponding entries, you obtain the following sum of four matrices Now try Exercise Example Using the Distributive Property Most graphing utilities have the capability of performing matrix operations Consult the user s guide for your graphing utility for specific keystrokes Try using a graphing utility to find the sum of the matrices and Technology A B Perform the indicated matrix operations Solution Now try Exercise In Example, you could add the two matrices first and then multiply the matrix by, as follows Notice that you obtain the same result

21 One important property of addition of real numbers is that the number is the additive identity That is, c c for any real number c For matrices, a similar property holds That is, if A is an m n matrix and O is the m n zero matrix consisting entirely of zeros, then A O A In other words, O is the additive identity for the set of all m n matrices For example, the following matrices are the additive identities for the set of all and matrices O Section Operations with Matrices 9 and O Remember that matrices are denoted by capital letters So, when you solve for X, you are solving for a matrix that makes the matrix equation true zero matrix zero matrix The algebra of real numbers and the algebra of matrices have many similarities For example, compare the following solutions Real Numbers (Solve for x) x a b x a a b a x b a x b a m n Matrices (Solve for X) X A B X A A B A X O B A X B A The algebra of real numbers and the algebra of matrices also have important differences, which will be discussed later Example Solving a Matrix Equation Solve for X in the equation X A B, where A and B Solution Begin by solving the equation for X to obtain X B A X B A Now, using the matrices A and B, you have X Substitute the matrices Subtract matrix A from matrix B Multiply the matrix by Now try Exercise

22 9 Chapter Matrices and Determinants Matrix Multiplication The third basic matrix operation is matrix multiplication At first glance, the definition may seem unusual You will see later, however, that this definition of the product of two matrices has many practical applications Definition of Matrix Multiplication If A a ij is an m n matrix and B b ij is an n p matrix, the product AB is an m p matrix AB c ij where c ij a i b j a i b j a i b j a in b nj 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 b b b a in b n a mn The definition of matrix multiplication indicates a row-by-column multiplication, where the entry in the ith row and jth column of the product AB is obtained by multiplying the entries in the ith row of A by the corresponding entries in the jth column of B and then adding the results The general pattern for matrix multiplication is as follows 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 First, note that the product AB is defined because the number of columns of A is equal to the number of rows of B Moreover, the product AB has order To find the entries of the product, multiply each row of A by each column of B, as follows AB 9 Now try Exercise 9

23 Section Operations with Matrices 9 Be sure you understand that for the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix 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 Exploration Use the following matrices to find AB, BA, ABC, and ABC What do your results tell you about matrix multiplication, commutativity, and associativity? A B C,, Example Find the product AB where A Finding the Product of Two Matrices 7 and Solution Note that the order of A is and the order of B is So, the product AB has order AB Now try Exercise B Example 9 Patterns in Matrix Multiplication a b 9 c The product AB for the following matrices is not defined A and B Now try Exercise

24 9 Chapter Matrices and Determinants Example Patterns in Matrix Multiplication a b Now try Exercise In Example, note that the two products are different Even if AB and BA are defined, matrix multiplication is not, in general, commutative That is, for most matrices, AB BA This is one way in which the algebra of real numbers and the algebra of matrices differ Properties of Matrix Multiplication Let A, B, and C be matrices and let c be a scalar ABC ABC Associative Property of Multiplication AB C AB AC Distributive Property A B)C AC BC Distributive Property cab cab AcB Associative Property of Scalar Multiplication Definition of Identity Matrix The n n matrix that consists of s on its main diagonal and s elsewhere is called the identity matrix of order n and is denoted by I n Identity matrix Note that an identity matrix must be square When the order is understood to be n, you can denote simply by I I n If A is an n n matrix, the identity matrix has the property that AI n A and I n A A For example, and AI A IA A

25 Applications Matrix multiplication can be used to represent a system of linear equations Note how the system ax a x a x b a x a x a x b a x a x a x b can be written as the matrix equation AX B, where A is the coefficient matrix of the system, and X and B are column matrices b b x x a a a a a a a a a x b A X B Section Operations with Matrices 9 Example Solving a System of Linear Equations The notation A B represents the augmented matrix formed when matrix B is adjoined to matrix A The notation I X represents the reduced rowechelon form of the augmented matrix that yields the solution to the system Consider the following system of linear equations x x x x x x x x a Write this system as a matrix equation, AX B b Use Gauss-Jordan elimination on the augmented matrix A B to solve for the matrix X Solution a In matrix form, AX B, the system can be written as follows x x b The augmented matrix is formed by adjoining matrix B to matrix A A B Using Gauss-Jordan elimination, you can rewrite this equation as I X So, the solution of the system of linear equations is x, x, and x, and the solution of the matrix equation is X x x x x Now try Exercise

26 9 Chapter Matrices and Determinants Example Softball Team Expenses Two softball teams submit equipment lists to their sponsors Women s Team Men s Team Bats Balls Gloves 7 Each bat costs $, 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 matrix form as and E 7 C The total cost of equipment for each team is given by the product CE 7 7 So, the total cost of equipment for the women s team is $ and the total cost of equipment for the men s team is $ Notice that you 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) Now try Exercise W RITING ABOUT MATHEMATICS Problem Posing Write a matrix multiplication application problem that uses the matrix A 7 Exchange problems with another student in your class Form the matrices that represent the problem, and solve the problem Interpret your solution in the context of the problem Check with the creator of the problem to see if you are correct Discuss other ways to represent and/or approach the problem

27 Section Operations with Matrices 97 Exercises VOCABULARY CHECK: In Exercises, fill in the blanks Two matrices are if all of their corresponding entries are equal When performing matrix operations, real numbers are often referred to as A matrix consisting entirely of zeros is called a matrix and is denoted by The n n matrix consisting of s on its main diagonal and s elsewhere is called the matrix of order n In Exercises and, match the matrix property with the correct form A, B, and C are matrices of order m n, and c and d are scalars (a) A A (i) Distributive Property (b) A B C A B C (ii) Commutative Property of Matrix Addition (c) c da ca da (iii) Scalar Identity Property (d) cda cda (iv) Associative Property of Matrix Addition (e) A B B A (v) Associative Property of Scalar Multiplication (a) A O A (i) Distributive Property (b) cab AcB (ii) Additive Identity of Matrix Addition (c) AB C AB AC (iii) Associative Property of Multiplication (d) ABC ABC (iv) Associative Property of Scalar Multiplication In Exercises, find x and y x 7 y 7 x y In Exercises, if possible, find (a) A B, (b) A B, (c) A, and (d) A B 7 x 7 A A A A y,,, x y x 7 B B B, B x y x 9 A B A A A, In Exercises, evaluate the expression 9 7 B,,, 7 B B 9 7 7

28 9 Chapter Matrices and Determinants In Exercises 9, use the matrix capabilities of a graphing utility to evaluate the expression Round your results to three decimal places, if necessary 9 X A B X A B X A B A B X 9 7 A A A A In Exercises 7, if possible, find AB and state the order of the result 7 B A 7, B A 7,, 7,, 7, B B B B In Exercises, solve fo X in the equation, given and B [ [ ] A ] In Exercises, use the matrix capabilities of a graphing utility to find AB, if possible B A 9, B A 7 9 In Exercises, if possible, find (a) AB, (b) BA, and (c) A (Note: A AA ) A A, A, A, A, A, 7 A, In Exercises 7, evaluate the expression Use the matrix capabilities of a graphing utility to verify your answer 7 A A A 9 B A, A, B, 9, 9, 7 B 7 B B B B B B 7 B 7,, B B

29 Section Operations with Matrices In Exercises, (a) write the system of linear equations as a matrix equation, AX B, and (b) use Gauss-Jordan elimination on the augmented matrix [A B] to solve for the matrix X 7 9 Manufacturing A corporation has three factories, each of which manufactures acoustic guitars and electric guitars The number of units of guitars produced at factory j in one day is represented by in the matrix A 7 Find the production levels if production is increased by % Manufacturing A corporation has four factories, each of which manufactures sport utility vehicles and pickup trucks The number of units of vehicle i produced at factory j in one day is represented by in the matrix A x x x x x x x x x x x x x x x x 9 x x x x x x x x x x x x 9x x x x x x x x 9 x x x x x x 7 9 x x x x x a ij Find the production levels if production is increased by % a ij 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, respectively The numbers of bushels of peaches sent to the three outlets are, 7, and, respectively The profit per bushel for apples is $ and the profit per bushel for peaches is $ (a) Write a matrix A that represents the number of bushels of each crop i that are shipped to each outlet j State what each entry a ij of the matrix represents (b) Write a matrix B that represents the profit per bushel of each fruit State what each entry b ij of the matrix represents (c) Find the product BA and state what each entry of the matrix represents Revenue A manufacturer of electronics produces three models of portable CD players, which are shipped to two warehouses The number of units of model i that are shipped to warehouse j is represented by in the matrix A,,, The prices per unit are represented by the matrix B $9 $ $ Compute BA and interpret the result Inventory A company sells five models of computers through three retail outlets The inventories are represented by S Model S A B C D E Outlet The wholesale and retail prices are represented by T Price T Wholesale Retail $ $ $ $ $,,, $ $ $ $ $ Model Compute ST and interpret the result A B C D E a ij

30 Chapter Matrices and Determinants Voting Preferences The matrix P From R D I is called a stochastic matrix Each entry p ij i j represents the proportion of the voting population that changes from party i to party j, and p ii represents the proportion that remains loyal to the party from one election to the next Compute and interpret P Voting Preferences Use a graphing utility to find P, P, P, P, P 7, and P for the matrix given in Exercise Can you detect a pattern as P is raised to higher powers? Labor/Wage Requirements A company that manufactures boats has the following labor-hour and wage requirements Labor per boat Department Cutting Assembly Packaging hr S hr hr Wages per hour Plant A T $ $9 $ B To Department Boat size Compute ST and interpret the result 7 Profit At a local dairy mart, the numbers of gallons of skim milk, % milk, and whole milk sold over the weekend are represented by A Skim % Whole milk milk milk A 7 7 $ $ $7 9 hr hr hr R D I 7 hr Cutting Assembly Packaging Small Medium Large Friday Saturday Sunday The selling prices (in dollars per gallon) and the profits (in dollars per gallon) for the three types of milk sold by the dairy mart are represented by B Selling price B Profit (a) Compute AB and interpret the result (b) Find the dairy mart s total profit from milk sales for the weekend Profit At a convenience store, the numbers of gallons of 7-octane, 9-octane, and 9-octane gasoline sold over the weekend are represented by A Octane A The selling prices per gallon and the profits per gallon for the three grades of gasoline sold by the convenience store are represents by B Selling Profit price B 9 Octane (a) Compute AB and interpret the result (b) Find the convenience store s profit from gasoline sales for the weekend 9 Exercise The numbers of calories burned by individuals of different body weights performing different types of aerobic exercises for a -minute time period are shown in matrix A Calories burned -lb person A lb person 9 79 Skim milk % milk Whole milk Bicycling Jogging Walking Friday Saturday Sunday (a) A -pound person and a -pound person bicycled for minutes, jogged for minutes, and walked for minutes Organize the time spent exercising in a matrix B (b) Compute BA and interpret the result

31 Section Operations with Matrices Synthesis True or False? In Exercises 7 and 7, determine whether the statement is true or false Justify your answer 7 Two matrices can be added only if they have the same order 7 Model It 7 Health Care The health care plans offered this year by a local manufacturing plant are as follows For individuals, the comprehensive plan costs $9, the HMO standard plan costs $, and the HMO Plus plan costs $9 For families, the comprehensive plan costs $7, the HMO standard plan costs $77 and the HMO Plus plan costs $ The plant expects the costs of the plans to change next year as follows For individuals, the costs for the comprehensive, HMO standard, and HMO Plus plans will be $9, $, and $997, respectively For families, the costs for the comprehensive, HMO standard, and HMO Plus plans will be $99, $7, and $7, respectively (a) Organize the information using two matrices A and B, where A represents the health care plan costs for this year and B represents the health care plan costs for next year State what each entry of each matrix represents (b) Compute A B and interpret the result (c) The employees receive monthly paychecks 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 employees paycheck this year and next year (d) Suppose the costs of each plan instead increase by % next year Write a matrix that shows the new monthly payment Think About It In Exercises 7, let matrices A, B, C, and D be of orders,,, and, respectively Determine whether the matrices are of proper order to perform the operation(s) If so, give the order of the answer 7 A C 7 B C 7 AB 7 BC 77 BC D 7 CB D 79 DA B BC DA Think About It 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 necessarily equal to B Illustrate this using the following matrices Think About It 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 necessarily true that A O or B O Illustrate this using the following matrices Exploration Let A and B be unequal diagonal matrices of the same order (A diagonal matrix is a square matrix in which each entry 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 Exploration Let i and let and (a) Find A, A, and A Identify any similarities with i, i, and i (b) Find and identify B Skills Review In Exercises 9, solve the equation x x x x 7 x x x x x x 9 x x x 9 x x x In Exercises 9 9, solve the system of linear equations both graphically and algebraically A A A i,, i x y 9 x y 9 x y 7 x 7y 7 x y x y x y 9x y B B, B i C i

32 Chapter Matrices and Determinants The Inverse of a Square Matrix What you should learn Verify 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 systems of linear equations Why you should learn it You can use inverse matrices to model and solve real-life problems For instance, in Exercise 7 on page, an inverse matrix is used to find a linear model for the number of licensed drivers in the United States The Inverse of a Matrix This section further develops the algebra of matrices To begin, consider the real number equation ax b To solve this equation for x, multiply each side of the equation by a (provided that a ) ax b a ax a b x a b x a b The number a is called the multiplicative inverse of a because a a The definition of the multiplicative inverse of a matrix is similar Definition of the Inverse of a Square Matrix Let A be an n n matrix and let I n be the n n identity matrix If there exists a matrix such that A AA I n A A then A is called the inverse of A The symbol A is read A inverse Example The Inverse of a Matrix Show that B is the inverse of A, where A and B Jon Love/Getty Images Solution To show that B is the inverse of A, show that AB I BA, as follows AB BA As you can see, AB I BA This is an example of a square matrix that has an inverse Note that not all square matrices have an inverse Now try Exercise Recall that it is not always 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 Example, you need only to check that AB I

33 Finding Inverse Matrices Section The Inverse of a Square Matrix If a matrix A has an inverse, A is called invertible (or nonsingular); otherwise, A is called singular A nonsquare matrix 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 matrix at the bottom of page ) If, however, a matrix does have an inverse, that inverse is unique Example shows how to use a system of equations to find the inverse of a matrix Example Finding the Inverse of a Matrix Find the inverse of Solution To find the inverse of A, try to solve the matrix equation AX I for X A X I Equating corresponding entries, you obtain two systems of linear equations Linear system with two variables, x and x x x x x Linear system with two variables, x and x x x x x Solve the first system using elementary row operations to determine that x and x From the second system you can determine that x and x Therefore, the inverse of A is You can use matrix multiplication to check this result Check A x x x x x x x x X A AA A A x x x x Now try Exercise