Chapter 16.4, Problem 16WE

To determine

To find: the matrix that represents the number of ways a message can be sent from one point to another using at most two relays.

Answer to Problem 16WE

  $\left[\begin{array}{cccc}1& 1& 0& 2\\ 1& 1& 0& 2\\ 1& 1& 0& 2\\ 0& 0& 0& 0\end{array}\right]$

Explanation of Solution

Given information:

Use the communication network at the right for figure,

  

Calculation:

The matrix representation is of the network diagram,

  

Is,

  $X=\begin{array}{c}A\\ B\\ C\\ D\end{array}\left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 0& 1\\ 0& 1& 0& 1\\ 0& 0& 0& 0\end{array}\right]$

Here $X$ represents the number of direct communication networks, $X^2$ represents the number of communication network using 2 steps. So $X+X^2$ represents the number of communication networks using no more than 2 steps.

  $X^2=XX$

  $\begin{array}{c}{X}^2=\left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 0& 1\\ 0& 1& 0& 1\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 0& 1\\ 0& 1& 0& 1\\ 0& 0& 0& 0\end{array}\right]\\ =\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 0& 1\\ 1& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right]\end{array}$

Now,

  $\begin{array}{c}X+X^2=\left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 0& 1\\ 0& 1& 0& 1\\ 0& 0& 0& 0\end{array}\right]+\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 0& 1\\ 1& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right]\\ =\left[\begin{array}{cccc}1& 1& 0& 2\\ 1& 1& 0& 2\\ 1& 1& 0& 2\\ 0& 0& 0& 0\end{array}\right]\end{array}$

Therefore, the matrix that represents the number of ways a message can be sent from one point to another using at most two relay is,

  $\left[\begin{array}{cccc}1& 1& 0& 2\\ 1& 1& 0& 2\\ 1& 1& 0& 2\\ 0& 0& 0& 0\end{array}\right]$