College Algebra 9th txtbk

SECTION 7–3 Matrix Operations 467
2.4 2.8 3.9
21. c
1.6 0 4.2 d c 7 2.2 2.2
3.2 3.2 1 d 59. Find a, b, c, and d so that
c a b
22. c 10
3 4 7
c d d c 2 3
0 1 d c 1 2
3 4 d
d [20 10] 23. 4 c
20 2 9 5 d
60. Find x and y so that
24. 5 c 7 3 0 9
4 5 6 2 d c 3x 5 2y 3
d c
1 4x 6 y d c 7 2
7 2 d
61. Find w, x, y, and z so that
Find the products in Problems 25–38.
3 0
25. [5 3] c 4 c
3
26. [2 4] c
7 d
8 d
7 11 d c w x
y z d c 9 1
4 6 d
62. Find x and y so that
3
27. c 5
28. c d [2 1]
3 d [4 2] x 1
4 c
2 y d c 4y 4 12 5
d c
5 3x 4 14 d
1
2
29. [3 2 4] £ 2 § 30. [1 2 2] £ 1 §
In Problems 63 and 64, let a, b, and c be any nonzero real
3
1
numbers, and let
1
2
31. £ 2 § [3 2 4] 32. £ 1 § [1 2 2]
A c a b and I c 1 0
c a d 0 1 d
3
1
3 7
33. c 6 3
34. c
1 9 dc 4
2 5 dc1 3 d 1 d
63. If A 2 0, how are a, b, and c related? Use this relationship to
provide several examples of 2 2 matrices with no zero entries
whose square is the zero matrix.
35. c 5 1
36. c 2 7 1
dc4
4 6 dc2 0
3 8 d 3 1 0 5 d
64. If A 2 I, how are a, b, and c related? Use this relationship to
provide several examples of 2 2 matrices with no zero entries
whose square is the matrix I.
37. c 8 3
38. c 7 0
0 3 dc 9 2
5 3 dc2 0
0 6 d 4 1 d
Problems 65 and 66 refer to the matrices
Problems 39–56 refer to the following matrices.
A c a b and
A c 2 1 3 B c 3 1
c d d 1 1 d
0 4 2 d 2 5 d
65. If AB 0, how are a, b, c, and d related? Use this relationship
to provide several examples of 2 2 matrices A with no zero
entries that satisfy AB 0.
D £
1 0 2
3 2
C £ 4 3 1§
0 1 §
57. 0.4 0.6
and B c
0.2 0.8 d
2 3 5
1 2
Perform the indicated operations, if possible.
39. CA 40. AC 41. BA
42. AB 43. C 2 44. B 2
45. C DA 46. B AD 47. 0.2CD
48. 0.1DB 49. 2DB 5CD 50. 3BA 4AC
51. (1)AC 3DB 52. (2)BA 6CD
53. CDA 54. ACD
55. DBA 56. BAD
In Problems 57 and 58, use a graphing calculator to calculate B,
B 2 , B 3 , . . . and AB, AB 2 , AB 3 , . . . . Describe any patterns you
observe in each sequence of matrices.
0.9 0.1
58. and B c
0.3 0.7 d
66. If BA 0, how are a, b, c, and d related? Use this relationship
to provide several examples of 2 2 matrices A with no zero
entries that satisfy BA 0.
67. Find x and y so that
B c 1 1
1 3
c
2 2 dcx 1
3 2 d c y 7
y 6 d
68. Find x and y so that
c x 1
1 0 dc2 1
4 1 d c y y
2 1 d
69. Find a, b, c, and d so that
c 1 3
1 4 dca b
c d d c 6 5
7 7 d
70. Find a, b, c, and d so that
c 1 2
2 3 dca b
c d d c 1 0
3 2 d
71. A square matrix is a diagonal matrix if all elements not on the
principal diagonal are zero. So a 2 2 diagonal matrix has the
form
A c a 0
0 d d