   Chapter 10.2, Problem 19ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Let A = [ 1 1 2     1 0 1     2 1 0 ] . Find A2 and A3. Let G be the graph with vertices v1, v2, and v3 and with A as its adjacency matrix. Use the answers to part (a) to find the number of walks of length 2 from v1 to v3. Do not draw G to solve this problem. Examine the calculations you performed in answering part (a) to find five walks of length 2 from v3 to v3. Then draw G and find the walks by visual inspection.

To determine

(a)

Find A2 and A3.

Explanation

Given information:

Let A = .

Calculation:

We have A=.

The first matrix A has dimensions 2×2 and the second matrix A has dimension 2×2.

Since the inner dimensions are equal, we can multiply the matrices. Moreover, the product should be a 2×2 matrix.

Multiply the rows of the first matrix with the columns of the second matrix (adding the products of the corresponding elements):

A2=A=[ 1 1 2 1 0 1 2 1 0][ 1 1 2 1 0 1 2 1 0]

=[ 11+11+22 11+10+21 12+11+20 11+01+12 11+00+11 12+01+10 21+11+02 21+10+01 22+11+00]=[ 1+1+4 1+0+2 2+1+0 1+0+2 1+0+1 2+0+0 2+1+0 2+0+0 4+1+0]=[ 6 3 3 3 2 2 3 2 5]

The first matrix A2 has dimensions 2×2 and the second matrix A has dimension 2×2

To determine

(b)

Find the number of walks of length 2 from v1 to v3 and the number of walks of length 3 from v1 to v3. Do not draw G to solve the problem.

To determine

(c)

Examine the calculations you performed in answering part (a) to find five walks of length 2 from v3 to v3. Then draw G and find the walks by visual inspection.

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