9 Matrices and Determinants In this chapter we discuss matrices in more detail. In the first three sections we define and study some algebraic operations on matrices, including addition, multiplication, and inversion. The next three sections deal with the determinant of a matrix. In the last chapter we used row operations and Gauss Jordan elimination to solve systems of linear equations. Row operations play a prominent role in the development of several topics in this chapter. One consequence of our discussion will be the development of two additional methods for solving systems of linear equations: one method involves inverse matrices and the other determinants. Matrices are both a very ancient and a very current mathematical concept. References to matrices and systems of equations can be found in Chinese manuscripts dating back to around B.C. Over the years, mathematicians and scientists have found many applications of matrices. More recently, the advent of personal and large-scale computers has increased the use of matrices in a wide variety of applications. In 99 Dan Bricklin and Robert Frankston introduced VisiCalc, the first electronic spreadsheet program for personal computers. Simply put, a spreadsheet is a computer program that allows the user to enter and manipulate numbers, often using matrix notation and operations. Spreadsheets were initially used by businesses in areas such as budgeting, sales projections, and cost estimation. However, many other applications have begun to appear. For example, a scientist can use a spreadsheet to analyze the results of an experiment, or a teacher can use one to record and average grades. There are even spreadsheets that can be used to help compute an individual s income tax. SECTION 9- Matrices: Basic Operations Addition and Subtraction Multiplication of a Matrix by a Number Matrix Product In Section 8- we introduced basic matrix terminology and solved systems of equations by performing row operations on augmented coefficient matrices. Matrices have many other useful applications and possess an interesting mathematical structure in their own right. As we will see, matrix addition and multiplication are similar to real number addition and multiplication in many respects, but there are some important differences. To help you understand the similarities and the differences, you should review the basic properties of real number operations discussed in Section A-. Addition and Subtraction Before we can discuss arithmetic operations for matrices, we have to define equality for matrices. Two matrices are equal if they have the same size and their corresponding elements are equal. For example,
9- Matrices: Basic Operations u v a b d e x y c f w z if and only if a u d x b v e y c w f z The sum of two matrices of the same size is a matrix with elements that are the sums of the corresponding elements of the two given matrices. Addition is not defined for matrices of different sizes. EXAMPLE Matrix Addition (A) (B) a c b d w y x z (a w) (c y) (b x) (d z) Matched Problem Add: Graphing utilities can also be used to solve problems involving matrix operations. Figure illustrates the solution to Example B on a graphing calculator. Because we add two matrices by adding their corresponding elements, it follows from the properties of real numbers that matrices of the same size are commutative and associative relative to addition. That is, if A, B, and C are matrices of the same size, then FIGURE Addition on a graphing calculator. A B B A Commutative (A B) C A (B C) Associative A matrix with elements that are all s is called a zero matrix. For example, the following are zero matrices of different sizes: [Note: may be used to denote the zero matrix of any size.] The negative of a matrix M, denoted by M, is a matrix with elements that are the negatives of the elements in M. Thus, if M a c b d
9 Matrices and Determinants then M a c b d Note that M (M) (a zero matrix). If A and B are matrices of the same size, then we define subtraction as follows: A B A (B) Thus, to subtract matrix B from matrix A, we simply subtract corresponding elements. EXAMPLE Matrix Subtraction Matched Problem Subtract: [ ] [ ] Multiplication of a Matrix by a Number The product of a number k and a matrix M, denoted by km, is a matrix formed by multiplying each element of M by k. EXAMPLE Multiplication of a Matrix by a Number Matched Problem Find:... EXPLORE-DISCUSS Multiplication of two numbers can be interpreted as repeated addition if one of the numbers is a positive integer. That is,
9- Matrices: Basic Operations a a a a a a a a a a a a and so on. Discuss this interpretation for the product of an integer k and a matrix M. Use specific examples to illustrate your remarks. We now consider an application that uses various matrix operations. EXAMPLE Sales and Commissions Ms. Fong and Mr. Petris are salespeople for a new car agency that sells only two models. August was the last month for this year s models, and next year s models were introduced in September. Gross dollar sales for each month are given in the following matrices: Fong $, Petris $, AUGUST SALES Compact Luxury $, $ A SEPTEMBER SALES Compact Luxury $, $88, $8, $, B For example, Ms. Fong had $, in compact sales in August and Mr. Petris had $, in luxury car sales in September. (A) What were the combined dollar sales in August and September for each salesperson and each model? (B) What was the increase in dollar sales from August to September? (C) If both salespeople receive a % commission on gross dollar sales, compute the commission for each salesperson for each model sold in September. Solutions We use matrix addition for part A, matrix subtraction for part B, and multiplication of a matrix by a number for part C. (A) Compact A B $8, $, Luxury $, $, Fong Petris (B) B A $8, $8, $, Fong Petris (C).B Compact (.)($,) (.)($8,) $, $8, $, $,8 Luxury (.)($88,) (.)($,) Fong Petris
9 Matrices and Determinants Matched Problem Repeat Example with A $, $, $, and B $8, $, $, Example involved an agency with only two salespeople and two models. A more realistic problem might involve salespeople and models. Problems of this size are often solved with the aid of a spreadsheet on a personal computer. Figure illustrates a computer spreadsheet solution for Example. A B C D E F G Compact Luxury Compact Luxury Compact Luxury August Sales September Sales September Commissions Fong Petris $, $, Combined Sales $, $ $, $8, Sales Increases $88, $, $, $, $8, $,8 Fong Petris $8, $, $, $, $8, $8, $, $, FIGURE Matrix Product Now we are going to introduce a matrix multiplication that may at first seem rather strange. In spite of its apparent strangeness, this operation is well-founded in the general theory of matrices and, as we will see, is extremely useful in many practical problems. Historically, matrix multiplication was introduced by the English mathematician Arthur Cayley (8 89) in studies of linear equations and linear transformations. In Section 9-, you will see how matrix multiplication is central to the process of expressing systems of equations as matrix equations and to the process of solving matrix equations. Matrix equations and their solutions provide us with an alternate method of solving linear systems with the same number of variables as equations. We start by defining the product of two special matrices, a row matrix and a column matrix. DEFINITION Product of a Row Matrix and a Column Matrix The product of a n row matrix and an n column matrix is a matrix given by n a n a... a n b b. b n a b a b... a n b n
9- Matrices: Basic Operations Note that the number of elements in the row matrix and in the column matrix must be the same for the product to be defined. EXAMPLE Product of a Row Matrix and a Column Matrix [()() ()() ()()] [ ] [] Matched Problem? Refer to Example. The distinction between the real number and the matrix [] is a technical one, and it is common to see matrices written as real numbers without brackets. In the work that follows, we will frequently refer to matrices as real numbers and omit the brackets whenever it is convenient to do so. EXAMPLE Production Scheduling A factory produces a slalom water ski that requires labor-hours in the fabricating department and labor-hour in the finishing department. Fabricating personnel receive $ per hour, and finishing personnel receive $8 per hour. Total labor cost per ski is given by the product [()() ()(8)] [ 8] [8] or $8 per ski 8 Matched Problem If the factory in Example also produces a trick water ski that requires labor-hours in the fabricating department and. labor-hours in the finishing department, write a product between appropriate row and column matrices that gives the total labor cost for this ski. Compute the cost. We now use the product of a n row matrix and an n column matrix to extend the definition of matrix product to more general matrices.
9 Matrices and Determinants DEFINITION Matrix Product If A is an m p matrix and B is a p n matrix, then the matrix product of A and B, denoted AB, is an m n matrix whose element in the ith row and jth column is the real number obtained from the product of the ith row of A and the jth column of B. If the number of columns in A does not equal the number of rows in B, then the matrix product AB is not defined. It is important to check sizes before starting the multiplication process. If A is an a b matrix and B is a c d matrix, then if b c, the product AB will exist and will be an a d matrix (see Fig. ). If b c, then the product AB does not exist. FIGURE Must be the same (b c) a b c d Size of product (a d) The definition is not as complicated as it might first seem. An example should help clarify the process. For A A is, B is, and so AB is. To find the first row of AB, we take the product of the first row of A with every column of B and write each result as a real number, not a matrix. The second row of AB is computed in the same manner. The four products of row and column matrices used to produce the four elements in AB are shown in the dashed box below. These products are usually calculated mentally, or with the aid of a calculator, and need not be written out. The shaded portions highlight the steps involved in computing the element in the first row and second column of AB. and B 9
9- Matrices: Basic Operations EXAMPLE Matrix Product (A) (B) Product is not defined (D) (F) (C) (E) Matched Problem Find each product, if it is defined: (A) (C) (E) (F) (D) (B) Figure illustrates a graphing calculator solution to Example A. What would you expect to happen if you tried to solve Example B on a graphing calculator? In the arithmetic of real numbers it does not matter in which order we multiply; for example,. In matrix multiplication, however, it does make a difference. That is, AB does not always equal BA, even if both multiplications are defined and both products are the same size (see Examples C and D). Thus, FIGURE Multiplication on a graphing calculator. Matrix multiplication is not commutative. Also, AB may be zero with neither A nor B equal to zero (see Example D). Thus, The zero property does not hold for matrix multiplication.
8 9 Matrices and Determinants (See Section A- for a discussion of the zero property for real numbers.) Just as we used the familiar algebraic notation AB to represent the product of matrices A and B, we use the notation A for AA, the product of A with itself, A for AAA, and so on. EXPLORE-DISCUSS In addition to the commutative and zero properties, there are other significant differences between real number multiplication and matrix multiplication. (A) In real number multiplication, the only real number whose square is is the real number ( ). Find at least one matrix A with all elements nonzero such that A, where is the zero matrix. (B) In real number multiplication, the only nonzero real number that is equal to its square is the real number ( ). Find at least one matrix A with all elements nonzero such that A A. We will continue our discussion of properties of matrix multiplication later in this chapter. Now we consider an application of matrix multiplication. EXAMPLE 8 Labor Costs Let us combine the time requirements for slalom and trick water skis discussed in Example and Matched Problem into one matrix: Trick ski Slalom ski Labor-hours per ski Assembly department h h Finishing department. h h L Now suppose that the company has two manufacturing plants, X and Y, in different parts of the country and that the hourly rates for each department are given in the following matrix: Assembly department Finishing department Hourly wages Plant X $ $8 Plant Y $ $ H Since H and L are both matrices, we can take the product of H and L in either order and the result will be a matrix: HL 8 LH. 8. 8 88 8 8 8
9- Matrices: Basic Operations 9 How can we interpret the elements in these products? Let s begin with the product HL. The element 8 in the first row and first column of HL is the product of the first row matrix of H and the first column matrix of L: Plant X Plant Y () () 8 8 Trick Slalom Notice that $ is the labor cost for assembling a trick ski at the California plant and $8 is the labor cost for assembling a slalom ski at the Wisconsin plant. Although both numbers represent labor costs, it makes no sense to add them together. They do not pertain to the same type of ski or to the same plant. Thus, even though the product HL happens to be defined mathematically, it has no useful interpretation in this problem. Now let s consider the product LH. The element in the first row and first column of LH is given by the following product: Assembly Finishing. Assembly ().(8) 8 Finishing where $ is the labor cost for assembling a trick ski at plant X and $ is the labor cost for finishing a trick ski at plant X. Thus, the sum is the total labor cost for producing a trick ski at plant X. The other elements in LH also represent total labor costs, as indicated by the row and column labels shown below: LH Labor costs per ski Plant X $ $8 Plant Y $8 $8 Trick ski Slalom ski Matched Problem 8 Refer to Example 8. The company wants to know how many hours to schedule in each department in order to produce, trick skis and, slalom skis. These production requirements can be represented by either of the following matrices: Trick skis P, Slalom skis, Q,, Trick skis Slalom skis Using the labor-hour matrix L from Example 8, find PL or LQ, whichever has a meaningful interpretation for this problem, and label the rows and columns accordingly.
9 Matrices and Determinants CAUTION Example 8 and Problem 8 illustrate an important point about matrix multiplication. Even if you are using a graphing utility to perform the calculations in a matrix product, it is still necessary for you to know the definition of matrix multiplication so that you can interpret the results correctly. Answers to Matched Problems.... (A) (B) $8, $, (C) $8, $88, $8, $, $, $, $,8. 8.. (A) Not defined (B) (C) (D) (E). or $ 8 (F) 9 8 8. Assembly Finishing PL,, Labor hours EXERCISE 9- A Perform the indicated operations in Problems 8, if possible.... 8... 8 8 8 9 9 8 9....... 8. 8 8. 8. 9.. 9 8 9 9 9
9- Matrices: Basic Operations B Find the products in Problems 9. 9...... Problems refer to the following matrices. C A Perform the indicated operations, if possible.. AB 8. BA 9. AC. CA. A. B. C. AD. CD. ()DB. AC BD 8. BA CD 9. DA C. B AD. DAC. CDB. ADB. BAB. Find a, b, c, and d so that. Find w, x, y, and z so that. Find x and y so that 8 w y.. x 8. Find x and y so that x 8 B D a c x z x y x y 8 b d 9 y y 9 8 C 9. Find a, b, c, and d so that. Find w, x, y, and z so that. Find a, b, c, and d so that. Find w, x, y, and z so that w y w y a c x z a c x z A diagonal matrix is a matrix of the form A a b d b d 8 d where a and d are any real numbers; if a d, A is called the identity matrix. In Problems, determine whether the statement is true or false. If true, explain why. If false, give a counterexample.. If A and B are diagonal matrices, then A B is a diagonal matrix.. If A and B are diagonal matrices, then AB is a diagonal matrix.. If A and B are diagonal matrices, then AB BA.. If A and B are matrices, then AB BA.. If A and B are diagonal matrices, then A B B A. 8. If A and B are matrices, then A B B A. 9. The zero matrix is a diagonal matrix.. If A and B are diagonal matrices such that AB, then A or B.. If A is the identity matrix and B is any matrix, then AB BA B.. If A and B are diagonal matrices such that AB B and B, then A is the identity matrix.. If A is a diagonal matrix such that A A, then A is the identity matrix.. If A is a diagonal matrix such that A, then A.
9 Matrices and Determinants APPLICATIONS. Cost Analysis. A company with two different plants manufactures guitars and banjos. Its production costs for each instrument are given in the following matrices: Materials Labor $ $ Find (A B), the average cost of production for the two plants.. Cost Analysis. If both labor and materials at plant X in Problem are increased %, find (.A B), the new average cost of production for the two plants.. Markup. An import car dealer sells three models of a car. Current dealer invoice price (cost) and the retail price for the basic models and the indicated options are given in the following two matrices (where Air means air conditioning): Model A Model B Model C Model A Model B Model C Plant X Guitar Banjo Basic car $, $, $, Basic car $,9 $, $8, $ $8 A Dealer invoice price AM/FM Air radio $8 $ $8 $ $9 $ Retail price AM/FM Air radio $8 $88 $9 Plant Y Guitar Banjo $ $ $ $9 $ $ $ B Cruise control $8 $8 $9 $ $ $8 M We define the markup matrix to be N M (markup is the difference between the retail price and the dealer invoice price). Suppose the value of the dollar has had a sharp decline and the dealer invoice price is to have an across-theboard % increase next year. In order to stay competitive with domestic cars, the dealer increases the retail prices only %. Calculate a markup matrix for next year s models and the indicated options. (Compute results to the nearest dollar.) 8. Markup. Referring to Problem, what is the markup matrix resulting from a % increase in dealer invoice prices and an increase in retail prices of %? (Compute results to the nearest dollar.) 9. Labor Costs. A company with manufacturing plants located in different parts of the country has labor-hour and wage requirements for the manufacturing of three types of inflatable boats as given in the following two matrices: Cruise control N Labor-hours per boat Cutting Assembly department department. h. h. h M. h.9 h. h. h. h. h Hourly wages Plant I Plant II $8 $9 N $ $ $ $ (A) Find the labor costs for a one-person boat manufactured at plant I. (B) Find the labor costs for a four-person boat manufactured at plant II. (C) Discuss possible interpretations of the elements in the matrix products MN and NM. (D) If either of the products MN or NM has a meaningful interpretation, find the product and label its rows and columns.. Inventory Value. A personal computer retail company sells five different computer models through three stores located in a large metropolitan area. The inventory of each model on hand in each store is summarized in matrix M. Wholesale (W) and retail (R) values of each model computer are summarized in matrix N. A M N Model B C D W $ $, $,8 $, $, Packaging department Cutting department Assembly department Packaging department Store Store Store (A) What is the retail value of the inventory at store? (B) What is the wholesale value of the inventory at store? (C) Discuss possible interpretations of the elements in the matrix products MN and NM. (D) If either of the products MN or NM has a meaningful interpretation, find the product and label its rows and columns. (E) Discuss methods of matrix multiplication that can be used to find the total inventory of each model on hand E R $8 $,8 $, $, $,9 A B C D E One-person boat Two-person boat Four-person boat
9- Matrices: Basic Operations at all three stores. State the matrices that can be used, and perform the necessary operations. (F) Discuss methods of matrix multiplication that can be used to find the total inventory of all five models at each store. State the matrices that can be used, and perform the necessary operations.. Airfreight. A nationwide airfreight service has connecting flights between five cities, as illustrated in the figure. To represent this schedule in matrix form, we construct a incidence matrix A, where the rows represent the origins of each flight and the columns represent the destinations. We place a in the ith row and jth column of this matrix if there is a connecting flight from the ith city to the jth city and a otherwise. We also place s on the principal diagonal, because a connecting flight with the same origin and destination does not make sense. Atlanta Baltimore Denver Chicago El Paso A Now that the schedule has been represented in the mathematical form of a matrix, we can perform operations on this matrix to obtain information about the schedule. (A) Find A. What does the in row and column of A indicate about the schedule? What does the in row and column indicate about the schedule? In general, how would you interpret each element off the principal diagonal of A? [Hint: Examine the diagram for possible connections between the ith city and the jth city.] (B) Find A. What does the in row and column of A indicate about the schedule? What does the in row and column indicate about the schedule? In general, how would you interpret each element off the principal diagonal of A? (C) Compute A, A A, A A A,..., until you obtain a matrix with no zero elements (except possibly on the principal diagonal), and interpret.. Airfreight. Find the incidence matrix A for the flight schedule illustrated in the figure. Compute A, A A, A A A,..., until you obtain a matrix with no zero elements (except possibly on the principal diagonal), and interpret. Origin Destination Louisville Milwaukee Phoenix. Politics. In a local election, a group hired a public relations firm to promote its candidate in three ways: telephone, house calls, and letters. The cost per contact is given in matrix M: M The number of contacts of each type made in two adjacent cities is given in matrix N: Telephone N,, Cost per contact $.8 $. $. Oakland House call 8 Telephone House call Letter Letter, 8, Berkeley Oakland (A) Find the total amount spent in Berkeley. (B) Find the total amount spent in Oakland. (C) Discuss possible interpretations of the elements in the matrix products MN and NM. (D) If either of the products MN or NM has a meaningful interpretation, find the product and label its rows and columns. (E) Discuss methods of matrix multiplication that can be used to find the total number of telephone calls, house calls, and letters. State the matrices that can be used, and perform the necessary operations. (F) Discuss methods of matrix multiplication that can be used to find the total number of contacts in Berkeley and in Oakland. State the matrices that can be used, and perform the necessary operations.. Nutrition. A nutritionist for a cereal company blends two cereals in different mixes. The amounts of protein, carbohydrate, and fat (in grams per ounce) in each cereal are given by matrix M. The amounts of each cereal used in the three mixes are given by matrix N. Newark
9 Matrices and Determinants Cereal A g/oz M g/oz g/oz N Mix X oz oz Cereal B g/oz g/oz g/oz Mix Y oz oz Mix Z oz oz Protein Carbohydrate Fat Cereal A Cereal B (A) Find the amount of protein in mix X. (B) Find the amount of fat in mix Z. (C) Discuss possible interpretations of the elements in the matrix products MN and NM. (D) If either of the products MN or NM has a meaningful interpretation, find the product and label its rows and columns. SECTION 9- Inverse of a Square Matrix Identity Matrix for Multiplication Inverse of a Square Matrix Application: Cryptography In this section we introduce the identity matrix and the inverse of a square matrix. These matrix forms, along with matrix multiplication, are then used to solve some systems of equations written in matrix form in Section 9-. Identity Matrix for Multiplication We know that for any real number a ()a a() a The number is called the identity for real number multiplication. Does the set of all matrices of a given dimension have an identity element for multiplication? That is, if M is an arbitrary m n matrix, does M have an identity element I such that IM MI M? The answer in general is no. However, the set of all square matrices of order n (matrices with n rows and n columns) does have an identity. DEFINITION Identity Matrix The identity matrix for multiplication for the set of all square matrices of order n is the square matrix of order n, denoted by I, with s along the principal diagonal (from upper left corner to lower right corner) and s elsewhere. For example, and are the identity matrices for all square matrices of order and, respectively.