Exam (elaborations) TEST BANK FOR Elementary Linear Algebra with Applications 9th Edition By Kolman, Hill (Solution manual)

Exam (elaborations) TEST BANK FOR Elementary Linear Algebra with Applications 9th Edition By Kolman, Hill (Solution manual) Instructor’s Solutions Manual Elementary Linear Algebra with Applications Ninth Edition Bernard Kolman Drexel University David R. Hill Temple University Contents Preface iii 1 Linear Equations and Matrices 1 1.1 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Algebraic Properties of Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Special Types of Matrices and Partitioned Matrices . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Matrix Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.7 Computer Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.8 Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2 Solving Linear Systems 27 2.1 Echelon Form of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Solving Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Elementary Matrices; Finding A−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4 Equivalent Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.5 LU-Factorization (Optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 Determinants 37 3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Properties of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 Cofactor Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4 Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.5 Other Applications of Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4 Real Vector Spaces 45 4.1 Vectors in the Plane and in 3-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.4 Span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.5 Span and Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6 Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.7 Homogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.8 Coordinates and Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.9 Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ii CONTENTS Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5 Inner Product Spaces 71 5.1 Standard Inner Product on R2 and R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2 Cross Product in R3 (Optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.3 Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.4 Gram-Schmidt Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.5 Orthogonal Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.6 Least Squares (Optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6 Linear Transformations and Matrices 93 6.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2 Kernel and Range of a Linear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3 Matrix of a Linear Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.4 Vector Space of Matrices and Vector Space of Linear Transformations (Optional) . . . . . . . 99 6.5 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.6 Introduction to Homogeneous Coordinates (Optional) . . . . . . . . . . . . . . . . . . . . . . 103 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7 Eigenvalues and Eigenvectors 109 7.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.2 Diagonalization and Similar Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.3 Diagonalization of Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 8 Applications of Eigenvalues and Eigenvectors (Optional) 129 8.1 Stable Age Distribution in a Population; Markov Processes . . . . . . . . . . . . . . . . . . . 129 8.2 Spectral Decomposition and Singular Value Decomposition . . . . . . . . . . . . . . . . . . . 130 8.3 Dominant Eigenvalue and Principal Component Analysis . . . . . . . . . . . . . . . . . . . . 130 8.4 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8.5 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 8.6 Real Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 8.7 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 8.8 Quadric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10 MATLAB Exercises 137 Appendix B Complex Numbers 163 B.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 B.2 Complex Numbers in Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Chapter 1 Linear Equations and Matrices Section 1.1, p. 8 2. x = 1, y = 2, z = −2. 4. No solution. 6. x = 13 + 10t, y = −8 − 8t, t any real number. 8. Inconsistent; no solution. 10. x = 2, y = −1. 12. No solution. 14. x = −1, y = 2, z = −2. 16. (a) For example: s = 0, t = 0 is one answer. (b) For example: s = 3, t = 4 is one answer. (c) s = t2 . 18. Yes. The trivial solution is always a solution to a homogeneous system. 20. x = 1, y = 1, z = 4. 22. r = −3. 24. If x1 = s1, x2 = s2, . . . , xn = sn satisfy each equation of (2) in the original order, then those same numbers satisfy each equation of (2) when the equations are listed with one of the original ones interchanged, and conversely. 25. If x1 = s1, x2 = s2, . . . , xn = sn is a solution to (2), then the pth and qth equations are satisfied. That is, ap1s1 + · · · + apnsn = bp aq1s1 + · · · + aqnsn = bq. Thus, for any real number r, (ap1 + raq1)s1 + · · · + (apn + raqn)sn = bp + rbq. Then if the qth equation in (2) is replaced by the preceding equation, the values x1 = s1, x2 = s2, . . . , xn = sn are a solution to the new linear system since they satisfy each of the equations. 2 Chapter 1 26. (a) A unique point. (b) There are infinitely many points. (c) No points simultaneously lie in all three planes. 28. No points of intersection: C1 C2 C2 C1 One point of intersection: C1 C2 Two points of intersection: C1 C2 Infinitely many points of intersection: C1= C2 30. 20 tons of low-sulfur fuel, 20 tons of high-sulfur fuel. 32. 3.2 ounces of food A, 4.2 ounces of food B, and 2 ounces of food C. 34. (a) p(1) = a(1)2 + b(1) + c = a + b + c = −5 p(−1) = a(−1)2 + b(−1) + c = a − b + c = 1 p(2) = a(2)2 + b(2) + c = 4a + 2b + c = 7. (b) a = 5, b = −3, c = −7. Section 1.2, p. 19 2. (a) A = ! """""# 0 1 0 0 1 1 0 1 1 1 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 $ %%%%%& (b) A = ! """""# 0 1 1 1 1 1 0 1 0 0 1 1 0 1 0 1 0 1 0 0 1 0 0 0 0 $ %%%%%& . 4. a = 3, b = 1, c = 8, d = −2. 6. (a) C + E = E + C = ! # 5 −5 8 4 2 9 5 3 4 $ &. (b) Impossible. (c) ' 7 −7 0 1 ( . (d) ! # −9 3 −9 −12 −3 −15 −6 −3 −9 $ &. (e) ! # 0 10 −9 8 −1 −2 −5 −4 3 $ &. (f) Impossible. 8. (a) AT = ! # 1 2 2 1 3 4 $ &, (AT )T = ' 1 2 3 2 1 4 ( . (b) ! # 5 4 5 −5 2 3 8 9 4 $ &. (c) ' −6 10 11 17 ( . Section 1.3 3 (d) ' 0 −4 4 0 ( . (e) ! # 3 4 6 3 9 10 $ &. (f) ' 17 2 −16 6 ( . 10. Yes: 2 ' 1 0 0 1 ( + 1 ' 1 0 0 0 ( = ' 3 0 0 2 ( . 12. ! # ! − 1 −2 −3 −6 ! + 2 −3 −5 −2 ! − 4 $ &. 14. Because the edges can be traversed in either direction. 16. Let x = ! """# x1 x2 ... xn $ %%%& be an n-vector. Then x + 0 = ! """# x1 x2 ... xn $ %%%& + ! """# 0 0 ... 0 $ %%%& = ! """# x1 + 0 x2 + 0 ... xn + 0 $ %%%& = ! """# x1 x2 ... xn $ %%%& = x. 18. )n i=1 )m j=1 aij = (a11 + a12 + · · · + a1m) + (a21 + a22 + · · · + a2m) + · · · + (an1 + an2 + · · · + anm) = (a11 + a21 + · · · + an1) + (a12 + a22 + · · · + an2) + · · · + (a1m + a2m + · · · + anm) = )m j=1 )n i=1 aij . 19. (a) True. )n i=1 (ai + 1) = )n i=1 ai + )n i=1 1 = )n i=1 ai + n. (b) True. )n i=1 * + )m j=1 1 , - = )n i=1 m = mn. (c) True. ! # )n i=1 ai $ & ! # )m j=1 bj $ & = a1 )m j=1 bj + a2 )m j=1 bj + · · · + an )m j=1 bj = (a1 + a2 + · · · + an) )m j=1 bj = )n i=1 ai )m j=1 bj = )m j=1 . )n i=1 aibj / 20. “new salaries” = u + .08u = 1.08u. Section 1.3, p. 30 2. (a) 4. (b) 0. (c) 1. (d) 1. 4. x = 5. 4 Chapter 1 6. x = ±"2, y = ±3. 8. x = ±5. 10. x = 65 , y = 12 5 . 12. (a) Impossible. (b) ! # 0 −1 1 12 5 17 19 0 22 $ &. (c) ! # 15 −7 14 23 −5 29 13 −1 17 $ &. (d) ! # 8 8 14 13 13 9 $ &. (e) Impossible. 14. (a) ' 58 12 66 13 ( . (b) Same as (a). (c) ' 28 8 38 34 4 41 ( . (d) Same as (c). (e) ' 28 32 16 18 ( ; same. (f) ' −16 −8 −26 −30 0 −31 ( . 16. (a) 1. (b) −6. (c) 0 −3 0 1 1 . (d) ! #−1 4 2 −2 8 4 3 −12 −6 $ &. (e) 10. (f) ! # 9 0 −3 0 0 0 −3 0 1 $ &. (g) Impossible. 18. DI2 = I2D = D. 20. ' 0 0 0 0 ( . 22. (a) ! ""# 1 14 0 13 $ %%& . (b) ! ""# 0 18 3 13 $ %%& . 24. col1(AB) = 1 ! # 1 2 3 $ & + 3 ! #−2 4 0 $ & + 2 ! #−1 3 −2 $ &; col2(AB) = −1 ! # 1 2 3 $ & + 2 ! #−2 4 0 $ & + 4 ! #−1 3 −2 $ &. 26. (a) −5. (b) BAT 28. Let A = 0 aij 1 be m × p and B = 0 bij 1 be p × n. (a) Let the ith row of A consist entirely of zeros, so that aik = 0 for k = 1, 2, . . . , p. Then the (i, j) entry in AB is )p k=1 aikbkj = 0 for j = 1, 2, . . . , n. (b) Let the jth column of A consist entirely of zeros, so that akj = 0 for k = 1, 2, . . . , m. Then the (i, j) entry in BA is )m k=1 bikakj = 0 for i = 1, 2, . . . , m. 30. (a) ! ""# 2 3 −3 1 1 3 0 2 0 3 2 3 0 −4 0 0 0 1 1 1 $ %%& . (b) ! ""# 2 3 −3 1 1 3 0 2 0 3 2 3 0 −4 0 0 0 1 1 1 $ %%& ! """""# x1 x2 x3 x4 x5 $ %%%%%& = ! ""# 7 −2 3 5 $ %%& . Section 1.3 5 (c) ! ""# 2 3 −3 1 1 7 3 0 2 0 3 −2 2 3 0 −4 0 3 0 0 1 1 1 5 $ %%& 32. ' −2 3 1 −5 (' x1 x2 ( = ' 5 4 ( . 34. (a) 2x1 + x2 + 3x3 + 4x4 = 0 3x1 − x2 + 2x3 = 3 −2x1 + x2 − 4x3 + 3x4 = 2 (b) same as (a). 36. (a) x1 ' 3 1 ( + x2 ' 2 −1 ( + x3 ' 1 4 ( = ' 4 −2 ( . (b) x1 ! #−1 2 3 $ & + x2 ! # 1 −1 1 $ & = ! # 3 −2 1 $ &. 38. (a) ' 1 2 0 2 5 3 (! # x1 x2 x3 $ & = ' 1 1 ( . (b) ! # 1 2 1 1 1 2 2 0 2 $ & ! # x1 x2 x3 $ & = ! # 0 0 0 $ &. 39. We have u · v = )n i=1 uivi = 0 u1 u2 · · · un 1 ! """# v1 v2 ... vn $ %%%& = uT v. 40. Possible answer: ! # 1 0 0 2 0 0 3 0 0 $ &. 42. (a) Can say nothing. (b) Can say nothing. 43. (a) Tr(cA) = )n i=1 caii = c )n i=1 aii = cTr(A). (b) Tr(A + B) = )n i=1 (aii + bii) = )n i=1 aii + )n i=1 bii = Tr(A) + Tr(B). (c) Let AB = C = 0 cij 1 . Then Tr(AB) = Tr(C) = )n i=1 cii = )n i=1 )n k=1 aikbki = )n k=1 )n i=1 bkiaik = Tr(BA). (d) Since aT ii = aii, Tr(AT) = )n i=1 aT ii = )n i=1 aii = Tr(A). (e) Let ATA = B = 0 bij 1 . Then bii = )n j=1 aT ijaji = )n j=1 a2 ji =$ Tr(B) = Tr(ATA) = )n i=1 bii = )n i=1 )n j=1 a2 ij % 0. Hence, Tr(ATA) % 0. 6 Chapter 1 44. (a) 4. (b) 1. (c) 3. 45. We have Tr(AB − BA) = Tr(AB) − Tr(BA) = 0, while Tr 2' 1 0 0 1 (3 = 2. 46. (a) Let A = 0 aij 1 and B = 0 bij 1 be m × n and n × p, respectively. Then bj = ! """# b1j b2j ... bnj $ %%%& and the ith entry of Abj is )n k=1 aikbkj, which is exactly the (i, j) entry of AB. (b) The ith row of AB is 04 k aikbk1 4 k aikbk2 · · · 4 k aikbkn 1 . Since ai = 0 ai1 ai2 · · · ain 1 , we have aib = 04 k aikbk1 4 k aikbk2 · · · 4 k aikbkn 1 . This is the same as the ith row of Ab. 47. Let A = 0 aij 1 and B = 0 bij 1 be m × n and n × p, respectively. Then the jth column of AB is (AB)j = ! "# a11b1j + · · · + a1nbnj ... am1b1j + · · · + amnbnj $ %& = b1j ! "# a11 ... am1 $ %& + · · · + bnj ! "# a1n ... amn $ %& = b1jCol1(A) + · · · + bnjColn(A). Thus the jth column of AB is a linear combination of the columns of A with coefficients the entries in bj . 48. The value of the inventory of the four types of items. 50. (a) row1(A) · col1(B) = 80(20) + 120(10) = 2800 grams of protein consumed daily by the males. (b) row2(A) · col2(B) = 100(20) + 200(20) = 6000 grams of fat consumed daily by the females. 51. (a) No. If x = (x1, x2, . . . , xn), then x · x = x21 + x22 + · · · + x2 n % 0. (b) x = 0. 52. Let a = (a1, a2, . . . , an), b = (b1, b2, . . . , bn), and c = (c1, c2, . . . , cn). Then (a) a · b = )n i=1 aibi and b · a = )n i=1 biai, so a · b = b · a. (b) (a + b) · c = )n i=1 (ai + bi)ci = )n i=1 aici + )n i=1 bici = a · c + b · c. (c) (ka) · b = )n i=1 (kai)bi = k )n i=1 aibi = k(a · b). Section 1.4 7 53. The i, ith element of the matrix AAT is )n k=1 aikaT ki = )n k=1 aikaik = )n k=1 (aik)2. Thus if AAT = O, then each sum of squares )n k=1 (aik)2 equals zero, which implies aik = 0 for each i and k. Thus A = O. 54. AC = ' 17 2 22 18 3 23 ( . CA cannot be computed. 55. BTB will be 6 × 6 while BBT is 1 × 1. Section 1.4, p. 40 1. Let A = 0 aij 1 , B = 0 bij 1 , C = 0 cij 1 . Then the (i, j) entry of A + (B + C) is aij + (bij + cij) and that of (A + B) + C is (aij + bij) + cij . By the associative law for addition of real numbers, these two entries are equal. 2. For A = 0 aij 1 , let B = 0 −aij 1 . 4. Let A = 0 aij 1 , B = 0 bij 1 , C = 0 cij 1 . Then the (i, j) entry of (A+B)C is )n k=1 (aik + bik)ckj and that of AC +BC is )n k=1 aikckj + )n k=1 bikckj. By the distributive and additive associative laws for real numbers, these two expressions for the (i, j) entry are equal. 6. Let A = 0 aij 1 , where aii = k and aij = 0 if i &= j, and let B = 0 bij 1 . Then, if i &= j, the (i, j) entry of AB is )n s=1 aisbsj = kbij , while if i = j, the (i, i) entry of AB is )n s=1 aisbsi = kbii. Therefore AB = kB. 7. Let A = 0 aij 1 and C = 0 c1 c2 · · · cm 1 . Then CA is a 1 × n matrix whose ith entry is )n j=1 cjaij . Since Aj = ! """# a1j a2j ... amj $ %%%& , the ith entry of )n j=1 cjAj is )m j=1 cjaij . 8. (a) ' cos 2" sin 2" −sin 2" cos 2" ( . (b) ' cos 3" sin 3" −sin 3" cos 3" ( . (c) ' cos k" sin k" −sin k" cos k" ( . (d) The result is true for p = 2 and 3 as shown in parts (a) and (b). Assume that it is true for p = k. Then Ak+1 = AkA = ' cos k" sin k" −sin k" cos k" (' cos " sin " −sin " cos " ( = ' cos k" cos " − sin k" sin " cos k" sin " + sin k" cos " −sin k" cos " − cos k" sin " cos k" cos " − sin k" sin " ( = ' cos(k + 1)" sin(k + 1)" −sin(k + 1)" cos(k + 1)" ( . Hence, it is true for all positive integers k. 8 Chapter 1 10. Possible answers: A = ' 1 0 0 1 ( ; A = ' 0 1 1 0 ( ; A = ! # 1 "2 1 "2 1 "2 − 1 "2 $ &. 12. Possible answers: A = ' 1 1 −1 −1 ( ; A = ' 0 0 0 0 ( ; A = ' 0 1 0 0 ( . 13. Let A = 0 aij 1 . The (i, j) entry of r(sA) is r(saij ), which equals (rs)aij and s(raij ). 14. Let A = 0 aij 1 . The (i, j) entry of (r + s)A is (r + s)aij , which equals raij + saij , the (i, j) entry of rA + sA. 16. Let A = 0 aij 1 , and B = 0 bij 1 . Then r(aij + bij) = raij + rbij . 18. Let A = 0 aij 1 and B = 0 bij 1 . The (i, j) entry of A(rB) is )n k=1 aik(rbkj), which equals r )n k=1 aikbkj, the (i, j) entry of r(AB). 20. 16 A, k = 16 . 22. 3. 24. If Ax = rx and y = sx, then Ay = A(sx) = s(Ax) = s(rx) = r(sx) = ry. 26. The (i, j) entry of (AT )T is the (j, i) entry of AT , which is the (i, j) entry of A. 27. (b) The (i, j) entry of (A + B)T is the (j, i) entry of 0 aij + bij 1 , which is to say, aji + bji. (d) Let A = 0 aij 1 and let bij = aji. Then the (i, j) entry of (cA)T is the (j, i) entry of 0 caij 1 , which is to say, cbij . 28. (A + B)T = ! # 5 0 5 2 1 2 $ &, (rA)T = ! # −4 −8 −12 −4 −8 12 $ &. 30. (a) ! #−34 17 −51 $ &. (b) ! #−34 17 −51 $ &. (c) BTC is a real number (a 1 × 1 matrix). 32. Possible answers: A = ' 1 −3 0 0 ( ; B = ' 1 2 23 1 ( ; C = ' −1 2 0 1 ( . A = ' 2 0 3 0 ( ; B = ' 0 0 1 0 ( ; C = ' 0 0 0 1 ( . 33. The (i, j) entry of cA is caij , which is 0 for all i and j only if c = 0 or aij = 0 for all i and j. 34. Let A = ' a b c d ( be such that AB = BA for any 2 × 2 matrix B. Then in particular, ' a b c d (' 1 0 0 0 ( = ' 1 0 0 0 (' a b c d ( ' a 0 c 0 ( = ' a b 0 0 ( so b = c = 0, A = ' a 0 0 d ( . Section 1.5 9 Also ' a 0 0 d (' 1 1 0 0 ( = ' 1 1 0 0 (' a 0 0