7 Engineering Example 1

7.1 Communication network

Problem in words

Figure 3 represents a communication network. Vertices $a,b,f$ and $g$ represent offices. Vertices $c,d$ and $e$ represent switching centres. The numbers marked along the edges represent the number of connections between any two vertices. Calculate the number of routes from $a$ and $b$ to $f$ and $g$

Figure 3 :

Mathematical statement of the problem

The number of routes from $a$ to $f$ can be calculated by taking the number via $c$ plus the number via $d$ plus the number via $e$ . In each case this is given by multiplying the number of connections along the edges connecting $a$ to $c$ , $c$ to $f$ etc. This gives the result:

Number of routes from $a$ to $f=3\times 2+4\times 6+1\times 1=31.$

The nature of matrix multiplication means that the number of routes is obtained by multiplying the matrix representing the number of connections from $ab$ to $cde$ by the matrix representing the number of connections from $cde$ to $fg$ .

Mathematical analysis

The matrix representing the number of routes from $ab$ to $cde$ is:

The matrix representing the number of routes from $cde$ to $fg$ is:

The product of these two matrices gives the total number of routes.

$\left(\begin{array}{ccc}\hfill 3\hfill & \hfill 4\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 3\hfill \end{array}\right)\left(\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 6\hfill & \hfill 3\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right)=\left(\begin{array}{ccc}\hfill 3\times 2+4\times 6+1\times 1\hfill & \hfill \hfill & \hfill 3\times 1+4\times 3+1\times 2\hfill \\ \hfill 2\times 2+1\times 6+3\times 1\hfill & \hfill \hfill & \hfill 2\times 1+1\times 3+3\times 2\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill 31\hfill & \hfill 17\hfill \\ \hfill 13\hfill & \hfill 11\hfill \end{array}\right)$

Interpretation

We can interpret the resulting (product) matrix by labelling the columns and rows.

Hence there are 31 routes from $a$ to $f$ , 17 from $a$ to $g$ , 13 from $b$ to $f$ and 11 from $b$ to $g$ .

Exercises

1. If $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right]B=\left[\begin{array}{cc}\hfill 5\hfill & \hfill 6\hfill \\ \hfill 7\hfill & \hfill 8\hfill \end{array}\right]C=\left[\begin{array}{cc}\hfill 0& \hfill -1\\ \hfill 2& \hfill -3\end{array}\right]$ find
   1. $AB$ ,
   2. $AC$ ,
   3. $\left(A+B\right)C$ ,
   4. $AC+BC$ (e)   $2A-3C$
2. If a rotation through an angle $\theta$ is represented by the matrix $A=\left[\begin{array}{cc}\hfill cos\theta & \hfill sin\theta \\ \hfill -sin\theta & \hfill cos\theta \end{array}\right]$ and a second rotation through an angle $\varphi$ is represented by the matrix $B=\left[\begin{array}{cc}\hfill cos\varphi & \hfill sin\varphi \\ \hfill -sin\varphi & \hfill cos\varphi \end{array}\right]$ show that both $AB$ and $BA$ represent a rotation through an angle $\theta +\varphi$ .
3. If $A=\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 3\\ \hfill -1& \hfill -1& \hfill -1\\ \hfill 2& \hfill 2& \hfill 2\end{array}\right],B=\left[\begin{array}{cc}\hfill 2& \hfill 4\\ \hfill -1& \hfill 2\\ \hfill 5& \hfill 6\end{array}\right],C=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right]$ , find $AB$ and $BC$ .
4. If $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 2\hfill \end{array}\right],B=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 5\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill -1\hfill \end{array}\right],C=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \\ \hfill -2\hfill \end{array}\right],$

   verify $A\left(BC\right)=\left(AB\right)C$ .

5. If $A=\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 3\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \end{array}\right]$ then show that $A{A}^T$ is symmetric.
6. If $A=\left[\begin{array}{cc}\hfill 11\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 1\hfill \end{array}\right]B=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 3\hfill \end{array}\right]$ verify that ${\left(AB\right)}^T=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 11\hfill & \hfill 3\hfill \\ \hfill 22\hfill & \hfill 7\hfill \end{array}\right]=B^T{A}^T$

Answer