Page 217 - Advanced engineering mathematics

7.1 Matrices 197

Suppose we want the number of walks of length 3 in this graph. Calculate
⎛ ⎞
0 5 1 4 2 4 3 2
⎜ 6 2 7 4 5 4 9 8 ⎟
⎜ ⎟
⎜ 1 7 0 8 3 2 3 2 ⎟
⎜ ⎟
4 4 8 6 8 8 11 10 ⎟
3
⎜
A = ⎜ ⎟ .
⎜ 2 5 3 8 4 6 8 4 ⎟
⎜ ⎟
⎜ 4 4 2 8 6 2 4 4 ⎟
⎜ ⎟
⎝ 3931184 6 7 ⎠
2821044 7 4
3
For example, the 4,7 element of A is 11, so there are 11 walks of length 3 between v 4 and v 7 .
There are no walks of length 3 between v 4 and v 6 .
Generally we would use a software package to compute A .
k
SECTION 7.1 PROBLEMS

In each of Problems 1 through 6, perform the requested −2 −4 6 8
8. A = ,B =
computation. 3 −1 1 −4
⎛ ⎞ ⎛ ⎞
1 −1 3 −4 0 0 ⎛ −3 ⎞
1. A= ⎝ 2 −4 6 ⎠ ,B= ⎝ −2 −1 6 ⎠ ;2A−3B
⎜ 2 ⎟
−1 1 2 8 15 4 9. A = −1 6 2 14 −22 ,B = ⎜ 6 ⎟

⎟
⎜
⎜ ⎟
⎝ 0 ⎠
⎛ ⎞ ⎛ ⎞
−2 2 4 4 −4
0 1 2 1
⎜ ⎟ ⎜ ⎟
2. A = ⎜ ⎟ ,B = ⎜ ⎟ ,−5A + 3B
⎝ 14 ⎝ 14
2 ⎠ 16 ⎠ ⎛ −3 1 ⎞
6 8 1 25
6 2 −16 0 0 28
⎜ ⎟
10. A = ⎜ ⎟ ,B =
⎝ 18 0 1 1 26
x 1 − x 1 −6
−22 ⎠
2
3. A = x ,B = ,A + 2AB 1 6
2 e x cos(x)
⎛ ⎞
4. A = (14),B = (−12),−3A − 5B −21 4 8 −3
⎜ 12 1 0 14 ⎟
1 −2 1 7 −9 11. A = ⎜ ⎟ ,
5. A = , ⎝ 1 16 0
8 2 −5 0 0 −8 ⎠
13 4 8 0

−5 1 8 21 7
B = ,4A + 8B
12 −6 −2 −1 9 −9 16 3 2
B =
5 9 14 0

−2 3 0 8
3
6. A = ,B = ,A − B 2
1 1 −5 1 −2 4 1 −3 7 2
12. A = ,B =
3 9 5 9 1 0
In each of Problems 7 through 16, determine which of AB
and BA are defined. Carry out all such products. ⎛ ⎞
−4 −2 0

⎛ ⎞ 13. A = ⎝ 0 5 3 ⎠ ,B = 1 −3 4
−4 6 2
7. A = ⎝ −2 −2 3 ⎠ , −3 1 1
1 1 8
3
⎛ ⎞
−2 4 6 12 5
⎛ ⎞
⎜ 0 ⎟
B = ⎝ −3 −3 1 1 4 ⎠ 14. A = ⎜ ⎟ ,B = 3 −2 4
⎝ −1 ⎠
0 0 1 6 −9
4
Copyright 2010 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.
October 14, 2010 14:23 THM/NEIL Page-197 27410_07_ch07_p187-246