Matrices.pdf

2 Matrices 2.1 Operations with Matrices 2.2 Properties of Matrix Operations 2.3 The Inverse of a Matrix...

2 Matrices 2.1 Operations with Matrices 2.2 Properties of Matrix Operations 2.3 The Inverse of a Matrix 2.4 Elementary Matrices 2.5 Markov Chains 2.6 More Applications of Matrix Operations Data Encryption (p. 94) Computational Fluid Dynamics (p. 79) Beam Deflection (p. 64) Information Retrieval (p. 58) Flight Crew Scheduling (p. 47) Clockwise from top left, Cousin_Avi/Shutterstock.com; Goncharuk/Shutterstock.com; 39 Gunnar Pippel/Shutterstock.com; Andresr/Shutterstock.com; nostal6ie/Shutterstock.com Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 40 Chapter 2 Matrices 2.1 Operations with Matrices Determine whether two matrices are equal. Add and subtract matrices and multiply a matrix by a scalar. Multiply two matrices. Use matrices to solve a system of linear equations. Partition a matrix and write a linear combination of column vectors. EQUALITY OF MATRICES In Section 1.2, you used matrices to solve systems of linear equations. This chapter introduces some fundamentals of matrix theory and further applications of matrices. It is standard mathematical convention to represent matrices in any one of the three ways listed below. 1. An uppercase letter such as A, B, or C 2. A representative element enclosed in brackets, such as [aij], [bij], or [cij] 3. A rectangular array of numbers [ ] a11 a12... a1n a21 a22... a2n ⋮ ⋮ ⋮ am1 am2... amn As mentioned in Chapter 1, the matrices in this text are primarily real matrices. That is, their entries are real numbers. Two matrices are equal when their corresponding entries are equal. Definition of Equality of Matrices Two matrices A = [aij] and B = [bij] are equal when they have the same size (m × n) and aij = bij for 1 ≤ i ≤ m and 1 ≤ j ≤ n. Equality of Matrices REMARK Consider the four matrices The phrase “if and only if” [13 ] , [1x ] means the statement is true in 2 2 A= , B= C = [1 3], and D=. both directions. For example, 4 4 “p if and only if q” means that p implies q and q implies p. Matrices A and B are not equal because they are of different sizes. Similarly, B and C are not equal. Matrices A and D are equal if and only if x = 3. A matrix that has only one column, such as matrix B in Example 1, is a column matrix or column vector. Similarly, a matrix that has only one row, such as matrix C in Example 1, is a row matrix or row vector. Boldface lowercase letters often designate column matrices and row matrices. For instance, matrix A in Example 1 can be [] [] 1 2 partitioned into the two column matrices a1 = and a2 = as shown below. 3 4 [13 ] [ ] 2 1 2 A= = = [a1 a2] 4 3 4 Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  2.1 Operations with Matrices 41 MATRIX ADDITION, SUBTRACTION, AND SCALAR MULTIPLICATION To add two matrices (of the same size), add their corresponding entries. Definition of Matrix Addition If A = [aij] and B = [bij] are matrices of size m × n, then their sum is the m × n matrix A + B = [aij + bij]. The sum of two matrices of different sizes is undefined. Addition of Matrices [−10 −1 + 1 2+3 ] [ ] [ ] [ ] 2 1 3 0 5 a. + = = 1 −1 2 0 + (−1) 1+2 −1 3 −2 −2 [01 ] [ ] [ ] 1 0 0 0 0 1 b. + = 2 3 0 0 0 1 2 3 REMARK [ ] [ ] [] 1 −1 0 [24 ] [ ] It is often convenient to rewrite 1 0 0 1 c. −3 + 3 = 0 d. + is undefined. the scalar multiple cA by 0 −1 −1 3 −2 2 0 factoring c out of every entry in the matrix. For example, factoring the scalar 12 out of When working with matrices, real numbers are referred to as scalars. To multiply the matrix below gives a matrix A by a scalar c, multiply each entry in A by c. [ ] 1 − 23 −3 2 5 2 1 2 = 12 [15 1. ] Definition of Scalar Multiplication If A = [aij] is an m × n matrix and c is a scalar, then the scalar multiple of A by c is the m × n matrix cA = [caij]. You can use −A to represent the scalar product (−1)A. If A and B are of the same size, then A − B represents the sum of A and (−1)B. That is, A − B = A + (−1)B. Scalar Multiplication and Matrix Subtraction For the matrices A and B, find (a) 3A, (b) −B, and (c) 3A − B. [ ] [ ] 1 2 4 2 0 0 A = −3 0 −1 and B= 1 −4 3 2 1 2 −1 3 2 SOLUTION [ ] [ ] [ ] 1 2 4 3(1) 3(2) 3(4) 3 6 12 a. 3A = 3 −3 0 −1 = 3(−3) 3(0) 3(−1) = −9 0 −3 2 1 2 3(2) 3(1) 3(2) 6 3 6 [ ] [ ] 2 0 0 −2 0 0 b. −B = (−1) 1 −4 3 = −1 4 −3 −1 3 2 1 −3 −2 [ ] [ ] [ ] 3 6 12 2 0 0 1 6 12 c. 3A − B = −9 0 −3 − 1 −4 3 = −10 4 −6 6 3 6 −1 3 2 7 0 4 Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 42 Chapter 2 Matrices MATRIX MULTIPLICATION Another basic matrix operation is matrix multiplication. To see the usefulness of this operation, consider the application below, in which matrices are helpful for organizing information. A football stadium has three concession areas, located in the south, north, and west stands. The top-selling items are peanuts, hot dogs, and soda. Sales for one day are given in the first matrix below, and the prices (in dollars) of the three items are given in the second matrix. Numbers of Items Sold Peanuts Hot Dogs Sodas Selling Price [ ][ ] South Stand 120 250 305 2.00 Peanuts North Stand 207 140 419 3.00 Hot Dogs West Stand 29 120 190 2.75 Soda To calculate the total sales of the three top-selling items at the south stand, multiply each entry in the first row of the matrix on the left by the corresponding entry in the price column matrix on the right and add the results. The south stand sales are (120)(2.00) + (250)(3.00) + (305)(2.75) = $1828.75 South stand sales Similarly, the sales for the other two stands are shown below. (207)(2.00) + (140)(3.00) + (419)(2.75) = $1986.25 North stand sales (29)(2.00) + (120)(3.00) + (190)(2.75) = $940.50 West stand sales The preceding computations are examples of matrix multiplication. You can write the product of the 3 × 3 matrix indicating the number of items sold and the 3 × 1 matrix indicating the selling prices as shown below. [ ][ ] [ ] 120 250 305 2.00 1828.75 207 140 419 3.00 = 1986.25 29 120 190 2.75 940.50 The product of these matrices is the 3 × 1 matrix giving the total sales for each of the three stands. The definition of the product of two matrices shown below is based on the ideas just developed. Although at first glance this definition may seem unusual, you will see that it has many practical applications. Definition of Matrix Multiplication If A = [aij] is an m × n matrix and B = [bij] is an n × p matrix, then the product AB is an m × p matrix AB = [cij] where n cij = ∑a b k=1 ik kj = ai1b1j + ai2b2j + ai3b3j +... + ainbnj. This definition means that to find the entry in the ith row and the jth column of the product AB, multiply the entries in the ith row of A by the corresponding entries in the jth column of B and then add the results. The next example illustrates this process. Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  2.1 Operations with Matrices 43 Finding the Product of Two Matrices Find the product AB, where [ ] −1 3 [−3 ] 2 A= 4 −2 and B=. −4 1 5 0 SOLUTION First, note that the product AB is defined because A has size 3 × 2 and B has size 2 × 2. Moreover, the product AB has size 3 × 2, and will take the form [ ][ [ ] −1 3 c11 c12 −3 ] 2 4 −2 = c21 c22. −4 1 5 0 c31 c32 To find c11 (the entry in the first row and first column of the product), multiply corresponding entries in the first row of A and the first column of B. That is, c11 = (−1)(−3) + (3)(−4) = −9 [ ][ [ ] −1 3 −9 c12 −3 ] 2 4 −2 = c21 c22. −4 1 5 0 c31 c32 Similarly, to find c12, multiply corresponding entries in the first row of A and the second column of B to obtain Arthur Cayley c12 = (−1)(2) + (3)(1) = 1 (1821–1895) [ ][ [ ] British mathematician −1 3 −9 1 −3 ] 2 Arthur Cayley is credited 4 −2 = c21 c22. with giving an abstract −4 1 5 0 c31 c32 definition of a matrix. Cayley was a Cambridge Continuing this pattern produces the results shown below. University graduate and a lawyer by profession. He c21 = (4)(−3) + (−2)(−4) = −4 began his groundbreaking c22 = (4)(2) + (−2)(1) = 6 work on matrices as he c31 = (5)(−3) + (0)(−4) = −15 studied the theory of c32 = (5)(2) + (0)(1) = 10 transformations. Cayley also was instrumental The product is in the development of [ ][ [ ] −1 3 −9 1 −3 ] determinants (discussed in 2 Chapter 3). Cayley and two AB = 4 −2 = −4 6. −4 1 American mathematicians, 5 0 −15 10 Benjamin Peirce (1809 –1880) and his Be sure you understand that for the product of two matrices to be defined, the son, Charles S. Peirce number of columns of the first matrix must equal the number of rows of the second (1839 –1914), are credited matrix. That is, with developing A B = AB. “matrix algebra.” m×n n×p m×p Equal Size of AB So, the product BA is not defined for matrices such as A and B in Example 4. Photo Researchers/Getty Images Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 44 Chapter 2 Matrices The general pattern for matrix multiplication is shown below. To obtain the element in the ith row and the jth column of the product AB, use the ith row of A and the jth column of B. [ ][ ][ ] a11 a12 a13... a1n c11 c12... c1j... c1p b11 b12... b1j... b1p a21 a22 a23... a2n c21 c22... c2j... c2p b21 b22... b2j... b2p ⋮ ⋮ ⋮ ⋮ b31 b32... b3j... b3p = ⋮ ⋮ ⋮ ⋮ ai1 ai2 ai3... ain ci1 ci2... cij... cip ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ bn1 bn2... bnj... bnp am1 am2 am3... amn cm1 cm2... cmj... cmp ai1b1j + ai2b2j + ai3b3j +... + ainbnj = cij DISCO VE RY Let [13 ] [01 ] 2 1 A= and B =. 4 2 1. Find A + B and B + A. Is matrix addition commutative? 2. Find AB and BA. Is matrix multiplication commutative? Matrix Multiplication See LarsonLinearAlgebra.com for an interactive version of this type of example. [ ] −2 4 2 −5 −1 [ ] [ ] 1 0 3 7 a. 1 0 0 = 2 −1 −2 −3 6 6 −1 1 −1 2×3 3×3 2×3 [−23 ][10 ] [ ] 4 0 3 4 b. = 5 1 −2 5 2×2 2×2 2×2 −1 [1 ][ ] [ ] 1 2 2 1 0 c. = 1 1 −1 0 1 2×2 2×2 2×2 [ ] 2 d. [1 −2 −3] −1 = 1 1×3 3×1 1×1 [ ] [ ] 2 2 −4 −6 e. −1 [1 −2 −3] = −1 2 3 1 1 −2 −3 3×1 1×3 3×3 Note the difference between the two products in parts (d) and (e) of Example 5. In general, matrix multiplication is not commutative. It is usually not true that the product AB is equal to the product BA. (See Section 2.2 for further discussion of the noncommutativity of matrix multiplication.) Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  2.1 Operations with Matrices 45 SYSTEMS OF LINEAR EQUATIONS One practical application of matrix multiplication is representing a system of linear equations. Note how the system a11x1 + a12x2 + a13x3 = b1 a21x1 + a22x2 + a23x3 = b2 a31x1 + a32x2 + a33x3 = b3 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. [ ][ ] [ ] a11 a12 a13 x1 b1 a21 a22 a23 x2 = b2 a31 a32 a33 x3 b3 A x = b Solving a System of Linear Equations Solve the matrix equation Ax = 0, where [] x1 −2 [ ]. 1 1 A= , x = x2 , and 0= 2 3 −2 x3 SOLUTION As a system of linear equations, Ax = 0 is x1 − 2x2 + x3 = 0 2x1 + 3x2 − 2x3 = 0. Using Gauss-Jordan elimination on the augmented matrix of this system, you obtain TECHNOLOGY Many graphing utilities and software programs can perform matrix addition, scalar [ 1 0 0 1 − 17 − 47 0 0 ]. multiplication, and matrix So, the system has infinitely many solutions. Here a convenient choice of a parameter multiplication. When you use is x3 = 7t, and you can write the solution set as a graphing utility to check one of the solutions in Example 6, x1 = t, x2 = 4t, x3 = 7t, t is any real number. you may see something similar In matrix terminology, you have found that the matrix equation to the screen below. [] x −2 [ ] [] 1 1 1 0 [A] x = [[1 -2 1 ] 2 3 −2 2 0 [2 3 -2]] x3 [B] [ has infinitely many solutions represented by [ ] [ ] [] ] x1 t 1 [A]*[B] [ x = x2 = 4t = t 4 , t is any scalar. ] x3 7t 7 The Technology Guide at That is, any scalar multiple of the column matrix on the right is a solution. Here are CengageBrain.com can help some sample solutions: you use technology to perform [] [ ] [] [ ] matrix operations. 1 2 0 −1 4 , 8 , 0 , and −4. 7 14 0 −7 Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 46 Chapter 2 Matrices PARTITIONED MATRICES The system Ax = b can be represented in a more convenient way by partitioning the matrices A and x in the manner shown below. If [ ] [] [] a11 a12... a1n x1 b1 a21 a22... a2n x2 b2 A= , x= , and b= ⋮ ⋮ ⋮ ⋮ ⋮ am1 am2... amn xn bm are the coefficient matrix, the column matrix of unknowns, and the right-hand side, respectively, of the m × n linear system Ax = b, then [ ][ ] a11 a12... a1n x1 a21 a22... a2n x2 =b ⋮ ⋮ ⋮ ⋮ am1 am2... amn xn [ ] a11x1 + a12 x2 +... + a1n xn a21x1 + a22 x2 +... + a2n xn =b ⋮ am1x1 + am2 x2 +... + amn xn [] [] [] a11 a12 a1n a21 a22 a2n x1 + x2 +... + xn = b. ⋮ ⋮ ⋮ am1 am2 amn In other words, Ax = x1a1 + x2a2 +... + xnan = b where a1, a2,... , an are the columns of the matrix A. The expression [] [] [] a11 a12 a1n a21 a22 a2n x1 + x2 +... + xn ⋮ ⋮ ⋮ am1 am2 amn is called a linear combination of the column matrices a1, a2,... , an with coefficients x1, x2,... , xn. Linear Combinations of Column Vectors The matrix product Ax is a linear combination of the column vectors a1, a2,... , an that form the coefficient matrix A. [] [] [] a11 a12 a1n a21 a22 a2n x1 + x2 +... + xn ⋮ ⋮ ⋮ am1 am2 amn Furthermore, the system Ax = b is consistent if and only if b can be expressed as such a linear combination, where the coefficients of the linear combination are a solution of the system. Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  2.1 Operations with Matrices 47 Solving a System of Linear Equations The linear system x1 + 2x2 + 3x3 = 0 4x1 + 5x2 + 6x3 = 3 7x1 + 8x2 + 9x3 = 6 can be rewritten as a matrix equation Ax = b, as shown below. [] [] [] [] 1 2 3 0 x1 4 + x2 5 + x3 6 = 3 7 8 9 6 Using Gaussian elimination, you can show that this system has infinitely many solutions, one of which is x1 = 1, x2 = 1, x3 = −1. [] [] [] [] 1 2 3 0 1 4 + 1 5 + (−1) 6 = 3 7 8 9 6 That is, b can be expressed as a linear combination of the columns of A. This representation of one column vector in terms of others is a fundamental theme of linear algebra. Just as you partition A into columns and x into rows, it is often useful to consider an m × n matrix partitioned into smaller matrices. For example, you can partition the matrix below as shown. [ ] [ ] 1 2 0 0 1 2 0 0 3 4 0 0 3 4 0 0 −1 −2 2 1 −1 −2 2 1 You can also partition the matrix into column matrices [ ] 1 2 0 0 3 4 0 0 = [c1 c2 c3 c4] −1 −2 2 1 or row matrices [ ] [] 1 2 0 0 r1 3 4 0 0 = r2. −1 −2 2 1 r3 Many real-life applications of linear systems involve LINEAR enormous numbers of equations and variables. For ALGEBRA example, a flight crew scheduling problem for American APPLIED Airlines required the manipulation of a matrix with 837 rows and more than 12,750,000 columns. To solve this application of linear programming, researchers partitioned the problem into smaller pieces and solved it on a computer. (Source: Very Large-Scale Linear Programming. A Case Study in Combining Interior Point and Simplex Methods, Bixby, Robert E., et al., Operations Research, 40, no. 5) Andresr/Shutterstock.com Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 48 Chapter 2 Matrices 2.1 Exercises See CalcChat.com for worked-out solutions to odd-numbered exercises. Equality of Matrices In Exercises 1–4, find x and y. Finding Products of Two Matrices In Exercises 15–28, find, if possible, (a) AB and (b) BA. x −2 −4 −2 1. [ = ] [ ] −1 7 y 7 22 15. A = [14 22], B = [−12 8] −5 x −5 [ ] [ ] 13 = 2 −2 16. A = [ 4] , B=[ −2] 2. 4 1 y 8 12 8 −1 2 [ ] [ ] 16 4 2x + 1 4 [ ] [ ] 16 4 5 4 2 −1 3 0 1 2 3. −3 13 15 6 = −3 13 15 3x 17. A = 5 1 −2 , B = −4 1 3 0 2 4 0 0 2 3y − 5 0 2 2 3 −4 −1 −2 [ ] [ ] x+2 −3 2x + 6 8 −3 [ ] [ ] 8 1 −1 7 1 1 2 4. 1 2y 2x = 1 18 −8 18. A = 2 −1 8 , B= 2 1 1 7 −2 y + 2 7 −2 11 3 1 −1 1 −3 2 [ ] ] [ Operations with Matrices In Exercises 5–10, find, if 2 1 0 −1 0 possible, (a) A + B, (b) A − B, (c) 2A, (d) 2A − B, and 19. A = −3 4 , B= 4 0 2 (e) B + 12A. 1 6 8 −1 7 [−34 −2 [12 ] ] [ ] [ ] 2 3 2 1 1 2 5. A = , B= 1 2 20. A = −3 0 4 , B= 2 −1 −2 −4 −2 [ ] [ ] 6 −1 1 4 4 1 6. A = 2 4 , B = −1 5 [] 2 −3 5 1 10 21. A = [3 2 1], B= 3 −3 [−12 ] [−32 ] 1 1 4 0 7. A = , B= −1 4 1 −2 [] −1 [ ] [ ] 3 2 −1 0 2 1 2 8. A = 2 B= 5 22. A = , B = [2 1 3 2] 4 5 , 4 2 −2 0 1 2 2 1 0 1 −1 [−16 ] [84 ] 0 3 [ ] 9. A = , B= −1 3 −4 −3 [10 ] 0 2 23. A = 4 −5 , B= 7 [ ] 3 0 2 10. A = 2 , B = [−4 6 2] [ ] 2 1 −1 −3 [ ] 2 24. A = , B= 1 3 11. Find (a) c21 and (b) c13, where C = 2A − 3B, 5 2 2 −1 2 −7 [ ] [ ] 5 4 4 1 [ ] [ ] A= , and B =. 0 −1 0 2 −3 1 2 0 −5 1 25. A = 4 0 2 , B = −3 12. Find (a) c23 and (b) c32, where C = 5A + 2B, 8 −1 7 1 [ ] [ ] 4 11 −9 1 0 5 [ ] [ ] 2 1 2 4 0 1 3 A= 0 3 2 , and B = −4 6 11. 26. A = 3 −1 −2 , B = −1 2 −3 −1 −3 1 1 −6 4 9 −2 1 −2 −2 1 4 3 13. Solve for x, y, and z in the matrix equation [] 6 [ ] [ ] [ ] x y y z 4 x 4 =2 +2. −2 z −1 −x 1 5 −x 27. A = , B = [10 12] 1 14. Solve for x, y, z, and w in the matrix equation 6 −4 [wy ] [ ] [ ] x 3 y w = +2. 3 −2 [16 ] [14 ] x 2 −1 z x 0 4 6 28. A = , B= 13 8 −17 20 2 Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.  2.1 Exercises 49 Matrix Size In Exercises 29–36, let A, B, C, D, and E Writing a Linear Combination In Exercises 49–52, be matrices with the sizes shown below. write the column matrix b as a linear combination of the A: 3 × 4 B: 3 × 4 C: 4 × 2 D: 4 × 2 E: 4 × 3 columns of A. −1 −1 [13 ] [ ] If defined, determine the size of the matrix. If not 2 49. A = , b= defined, explain why. −3 1 7 [ [] ] 29. A + B 30. C + E 1 2 4 1 1 31. 2D 32. −4A 50. A = −1 0 2 , b= 3 33. AC 34. BE 0 1 3 2 [ ] [] 35. E − 2A 36. 2D + C 1 1 −5 3 51. A = 1 0 −1 , b = 1 Solving a Matrix Equation In Exercises 37 and 38, 2 −1 −1 0 solve the matrix equation Ax = 0. [ ] [ ] −3 −22 [] x1 5 −1 −1 [ ] [] 2 0 52. A = 3 4 , b= 4 37. A = , x = x2 , 0 = 1 −2 2 0 4 −8 32 x3 [] x1 [ ] [] 1 2 1 3 0 Solving a Matrix Equation In Exercises 53 and 54, x solve for A. 38. A = 1 −1 0 1 , x= 2 , 0= 0 x3 −1 [13 ] [ ] 0 1 2 0 2 1 0 x4 53. A= 5 0 1 −1 54. [ −2] A=[ 1] Solving a System of Linear Equations In Exercises 2 1 0 39–48, write the system of linear equations in the form 3 0 Ax = b and solve this matrix equation for x. 39. −x1 + x2 = 4 40. 2x1 + 3x2 = 5 Solving a Matrix Equation In Exercises 55 and 56, −2x1 + x2 = 0 x1 + 4x2 = 10 solve the matrix equation for a, b, c, and d. [13 ] [ac ] [ ] 41. −2x1 − 3x2 = −4 42. −4x1 + 9x2 = −13 2 b 6 3 55. = 4 d 19 2 6x1 + x2 = −36 x1 − 3x2 = 12 56. [ d] [ 3 1] [ 4 ] 43. x1 − 2x2 + 3x3 = 9 a b 2 1 3 17 = c −1 −x1 + 3x2 − x3 = −6 2x1 − 5x2 + 5x3 = 17 Diagonal Matrix In Exercises 57 and 58, find the 44. x1 + x2 − 3x3 = −1 product AA for the diagonal matrix. A square matrix [ ] −x1 + 2x2 = 1 a11 0 0... 0 x1 − x2 + x3 = 2 0 a22 0... 0 45. x1 − 5x2 + 2x3 = −20 A= 0 0 a33... 0 −3x1 + x2 − x3 = 8 ⋮ ⋮ ⋮ ⋮ 0 0 0... ann −2x2 + 5x3 = −16 is a diagonal matrix when all entries that are not on the 46. x1 − x2 + 4x3 = 17 main diagonal are zero. x1 + 3x2 = −11 [