College Algebra 9th txtbk

SECTION 7–4 Solving Systems of Linear Equations Using Matrix Inverse Methods 485
47. c 3 9
2 4
48. c
2 6 d 3 6 d
66.
49. c 2 3
50. c 5 4
3 5 d 4 3 d
1 1 0
51. £ 1 1 1 § 52.
0 1 1
1 2 5
1 1 1
53. £ 3 5 9§
54. £ 2 3 2§
1 1 2
3 3 2
2 2 1
4 2 1
55. £ 0 4 1 § 56. £ 1 1 1 §
1 2 1
3 1 1
2 1 1
57. £ 1 1 0§
58.
1 1 0
£
1 5 10
1 5 10
59. £ 0 1 4§
60. 0 1 6§
61. x 1 2x 2 k 1
1 6 15
1 4 3
Write each system in Problems 61–68 as a matrix equation and
solve using inverses. [ Note: the inverse of each coefficient matrix
was found earlier in this exercise in the indicated problem. ]
2x 1 5x 2 k 2
(A)
(B)
(C)
k 1 2, k 2 5
k 1 4, k 2 1
k 1 3, k 2 2
(see Problem 43.)
62.
63.
64.
65.
3x 1 4x 2 k 1
2x 1 3x 2 k 2
(A)
(B)
(C)
(see Problem 44.)
k 1 3, k 2 1
k 1 6, k 2 5
k 1 0, k 2 4
5x 1 7x 2 k 1
2x 1 3x 2 k 2
(A)
(B)
(C)
(see Problem 45.)
k 1 5, k 2 1
k 1 8, k 2 4
k 1 6, k 2 0
11x 1 4x 2 k 1
3x 1 x 2 k 2
(A)
(B)
(C)
(see Problem 46.)
k 1 2, k 2 3
k 1 1, k 2 9
k 1 4, k 2 5
x 1 x 2 k 1
x 1 x 2 x 3 k 2
x 2 x 3 k 3
(A) k 1 1, k 2 1, k 3 2
(B) k 1 1, k 2 0, k 3 4
(C) k 1 3, k 2 2, k 3 0
(see Problem 51.)
2 1 0
£
0 1 1
1 0 1
1 1 0
£
2 1 1
0 1 1
§
§
67.
2x 1 x 2 k 1
x 2 x 3 k 2
x 1 x 3 k 3
(A)
(B)
(C)
(see Problem 52.)
k 1 2, k 2 4, k 3 1
k 1 2, k 2 3, k 3 1
k 1 1, k 2 2, k 3 5
x 1 2x 2 5x 3 k 1
3x 1 5x 2 9x 3 k 2
x 1 x 2 2x 3 k 3
(A)
(B)
(C)
(see Problem 53.)
k 1 0, k 2 1, k 3 4
k 1 5, k 2 1, k 3 0
k 1 6, k 2 0, k 3 2
68. x 1 x 2 x 3 k 1
2x 1 3x 2 2x 3 k 2
3x 1 3x 2 2x 3 k 3
(A) k 1 3, k 2 1, k 3 0
(B) k 1 0, k 2 4, k 3 5
(C) k 1 2, k 2 0, k 3 1
(see Problem 54.)
For n n matrices A and B and n 1 matrices C, D, and X,
solve each matrix equation in Problems 69–74 for X. Assume all
necessary inverses exist.
69. AX BX C 70. AX BX C D
71. X AX C
72. X C AX BX
73. AX C 3X 74. AX C BX 7X D
75. Discuss the existence of for 2 2 diagonal matrices of
the form
A c a 0
0 d d
A 1
76. Discuss the existence of for 2 2 upper triangular matrices
of the form
A 1
A c a b
0 d d
77. Find A 1 and A 2 for each of the following matrices.
(A) A c 3 2
(B) A c 2 1
4 3 d
3 2 d
78. Based on your observations in Problem 77, if A A 1 for a
square matrix A, what is A 2 ? Give a mathematical argument to
support your conclusion.
79. Find (A 1 ) 1 for each of the following matrices.
5 5
(A) A c 4 2 (B) A c
1 3 d
1 3 d
80. Based on your observations in Problem 79, if exists for a
square matrix A, what is (A 1 ) 1 ? Give a mathematical argument
to support your conclusion.
A 1
81. Find (AB) 1 , A 1 B 1 , and B 1 A 1 for each of the following
pairs of matrices.
(A) A c 3 4 and B c 3 7
2 3 d 2 5 d