Chapter 16.4, Problem 9ST

a.

To determine

To find: The communication matrix for the network shown.

Answer to Problem 9ST

The answer is shown below.

Explanation of Solution

Given:

  

Calculation:

Let $X$ be the matrix.

The arrows in the diagram shows the direction in which data can be transmitted and received. The rows of matrix represented the directions of the data transmitted and the columns represent the data to be received. Use $1$ if direct transmission is possible and $0$ if it is not possible.

The matrix is,

  $X=\left[\begin{array}{l}ABCD\\ 0111\\ 1010\\ 1000\\ 1010\end{array}\right]$ .

b.

To determine

To find: The number of roots a message can sent from point $A$ to $C$ using at most one relay point.

Answer to Problem 9ST

The answer is shown below.

Explanation of Solution

Given:

  

Calculation:

Total number of roots a message can be sent from one point to another using at most one relay is $X+X^2$ .

Now $X^2$ is,

  $\begin{array}{c}{X}^2=\left[\begin{array}{l}011\text{}1\\ 1010\\ 1000\\ 1010\end{array}\right]\left[\begin{array}{l}011\text{}1\\ 1010\\ 1000\\ 1010\end{array}\right]\\ =\left[\begin{array}{l}3020\\ 1111\\ 0111\\ 1111\end{array}\right]\end{array}$

Now $X+X^2$ is,

  $\begin{array}{c}X+X^2=\left[\begin{array}{l}011\text{}1\\ 1010\\ 1000\\ 1010\end{array}\right]+\left[\begin{array}{l}3020\\ 1111\\ 0111\\ 1111\end{array}\right]\\ =\left[\begin{array}{c}3131\\ 2121\\ 1111\\ 2121\end{array}\right]\end{array}$

Therefore, the above matric represents the number of roots a message can sent from point $A$ to $C$ using at most one relay.