1 Applications of matrices

The solution of simultaneous linear equations is a task frequently occurring in engineering. In electrical engineering the analysis of circuits provides a ready example.

However the simultaneous equations arise, we need to study two things:

1. how we can conveniently represent large systems of linear equations
2. how we might find the solution of such equations.

We shall discover that knowledge of the theory of matrices is an essential mathematical tool in this area.

1.1 Representing simultaneous linear equations

Suppose that we wish to solve the following three equations in three unknowns $x_1,x_2$ and $x_3$ :

We can isolate three facets of this system: the coefficients of $x_1,x_2,x_3$ ; the unknowns $x_1,x_2,x_3$ ; and the numbers on the right-hand sides.

Notice that in the system

the only difference from the first system is the names given to the unknowns. It can be checked that the first system has the solution $x_1=2,x_2=-1,x_3=1.$ The second system therefore has the solution $x=2,y=-1,z=1.$

We can isolate the three facets of the first system by using arrays of numbers and of unknowns:

$\left[\begin{array}{ccc}\hfill 3& \hfill 2& \hfill -1\\ \hfill 1& \hfill -1& \hfill 1\\ \hfill 2& \hfill 3& \hfill 4\end{array}\right]\left[\begin{array}{c}\hfill x_1\hfill \\ \hfill x_2\hfill \\ \hfill x_3\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 3\hfill \\ \hfill 4\hfill \\ \hfill 5\hfill \end{array}\right]$

Even more conveniently we represent the arrays with letters (usually capital letters)

$AX=B$

Here, to be explicit, we write

$A=\left[\begin{array}{ccc}\hfill 3& \hfill 2& \hfill -1\\ \hfill 1& \hfill -1& \hfill 1\\ \hfill 2& \hfill 3& \hfill 4\end{array}\right]X=\left[\begin{array}{c}\hfill x_1\hfill \\ \hfill x_2\hfill \\ \hfill x_3\hfill \end{array}\right]B=\left[\begin{array}{c}\hfill 3\hfill \\ \hfill 4\hfill \\ \hfill 5\hfill \end{array}\right]$

Here $A$ is called the matrix of coefficients , $X$ is called the matrix of unknowns and $B$ is called the matrix of constants .

If we now append to $A$ the column of right-hand sides we obtain the augmented matrix for the system:

$\left[\left.\begin{array}{ccc}\hfill 3& \hfill 2& \hfill -1\\ \hfill 1& \hfill -1& \hfill 1\\ \hfill 2& \hfill 3& \hfill 4\end{array}\right|\right.\left.\begin{array}{c}\hfill 3\hfill \\ \hfill 4\hfill \\ \hfill 5\hfill \end{array}\right]$

The order of the entries, or elements, is crucial. For example, all the entries in the second row relate to the second equation, the entries in column 1 are the coefficients of the unknown $x_1$ , and those in the last column are the constants on the right-hand sides of the equations.

In particular, the entry in row 2 column 3 is the coefficient of $x_3$ in equation 2.

1.2 Representing networks

Shortest-distance problems are important in communications study. Figure 1 illustrates schematically a system of four towns connected by a set of roads.

Figure 1

The system can be represented by the matrix

$abcd$

$\begin{array}{c}\hfill a\hfill \\ \hfill b\hfill \\ \hfill c\hfill \\ \hfill d\hfill \end{array}\left[\begin{array}{ccccccc}\hfill 0\hfill & \hfill \hfill & \hfill 1\hfill & \hfill \hfill & \hfill 0\hfill & \hfill \hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill \hfill & \hfill 0\hfill & \hfill \hfill & \hfill 1\hfill & \hfill \hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill 1\hfill & \hfill \hfill & \hfill 0\hfill & \hfill \hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill 1\hfill & \hfill \hfill & \hfill 1\hfill & \hfill \hfill & \hfill 0\hfill \end{array}\right]$

The row refers to the town from which the road starts and the column refers to the town where the road ends. An entry of 1 indicates that two towns are directly connected by a road (for example $b$ and $d$ ) and an entry of zero indicates that there is no direct road (for example $a$ and $c$ ). Of course, if there is a road from $b$ to $d$ (say) it is also a road from $d$ to $b$ .

In this Section we shall develop some basic ideas about matrices.