Page 218 - Advanced engineering mathematics

198 CHAPTER 7 Matrices and Linear Systems

7 −8 1 −4 3
15. A = ,B = v 2 G
1 6 −4 7 0
v 1
⎛ ⎞
−3 2
0 −2 v
⎜ ⎟ v 5 3
16. A = ⎜ ⎟ ,B = −5 5 7 2
⎝ 1
8 ⎠
v Problem 23
3 −3 4
In each of Problems 17 through 21, determine if AB and/or
v 1 H
BA is defined. For those products that are defined, give the
v
dimensions of the product matrix. 5
v 2
17. A is 14 × 21, B is 21 × 14. v 4 v 3 Problem 24
18. A is 18 × 4, B is 18 × 4.
v 1
19. A is 6 × 2, B is 4 × 6. K
v 5
20. A is 1 × 3, B is 3 × 3. v 2
21. A is 7 × 6, B is 7 × 7. v
v 4 3
22. Find nonzero 2 × 2 matrices A, B,and C such that Problem 25
BA = CA but B = C.
FIGURE 7.4 Graphs of
23. For the graph G of Figure 7.4, determine the number
Problems 23, 24, and 25,
of v 1 − v 4 walks of length 3, the number of v 2 − v 3
walks of length 3, and the number of v 2 − v 4 walks of in Section 7.1.
length 4.
3
24. For the graph H of Figure 7.4, determine the number (b) Prove that the i, j-element of A equals twice the
number of triangles in G containing v i as a vertex. A
of v 1 − v 4 walks of length 4 and the number of v 2 − v 3
walks of length 2. triangle in G consists of three points, each a neighbor
of the other.
25. For the graph K of Figure 7.4, determine the number
27. Show that the set of all n × m matrices with real ele-
of v 4 − v 5 walks of length 2, the number of v 2 − v 3
walks of length 3, and the number of v 1 − v 2 walks ments is a vector space, using the usual addition of
and v 4 − v 5 walks of length 4. matrices and multiplication of matrices by scalars as
the operations. What is the dimension of this vector
26. Let A be the adjacency matrix of a graph G.
2
(a) Prove that the i, j-element of A equals the number space?
of points of G that are neighbors of v i in G.This 28. Redo Problem 27 for the case that the elements in the
number is called the degree of v i . matrices are complex numbers.

7.2 Elementary Row Operations

Some applications, as well as determining certain information about matrices, make
use of elementary row operations. We will define three such operations. Let A be a
matrix.

1. Type I operation: interchange two rows of A.
2. Type II operation: multiply a row of A by a nonzero number.
3. Type III operation: add a scalar multiple of one row to another row of A.

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October 14, 2010 14:23 THM/NEIL Page-198 27410_07_ch07_p187-246