3 Non-unique solutions

The key to obtaining a unique solution of the system $AX=B$ is to find $A^{-1}$ . What happens when $A^{-1}$ does not exist?

Consider the system

$2x_1+3x_2=5$ (1)

$4x_1+6x_2=10$ (2)

In matrix form this becomes

$\left[\begin{array}{cc}\hfill 2\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 6\hfill \end{array}\right]\left[\begin{array}{c}\hfill x_1\hfill \\ \hfill x_2\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 5\\ \hfill 10\end{array}\right].$

Task!

Identify the matrix $A$ and hence find $A^{-1}$ .

Answer

Looking at the original system we see that Equation (2) is simply Equation (1) with all coefficients doubled. In effect we have only one equation for the two unknowns $x_1$ and $x_2$ . The equations are consistent and have infinitely many solutions .

If we let $x_2$ assume a particular value, $t$ say, then $x_1$ must take the value $x_1=\frac{1}{2}\left(5-3t\right)$ i.e. the solution to the given equations is:

$x_2=t,x_1=\frac{1}{2}\left(5-3t\right),\text{where}t\text{is called a parameter.}$

For each value of $t$ there are unique values for $x_1$ and $x_2$ which satisfy the original system of equations. For example, if $t=1$ , then $x_2=1$ , $x_1=1$ , if $t=-3$ then $x_2=-3$ , $x_1=7$ and so on.

Now consider the system

$2x_1+3x_2=5$ (3)

$4x_1+6x_2=9$ (4)

Since the left-hand sides are the same as those in the previous system then $A$ is the same and again $A^{-1}$ does not exist. There is no solution to the Equations (3) and (4).

However, if we double Equation (3) we obtain

$4x_1+6x_2=10,$

which conflicts with Equation (4). There are thus no solutions to (3) and (4) and the equations are said to be inconsistent .

Task!

What can you conclude about the solutions of the following systems?

1. $$\begin{array}{ccccc}\hfill x_1& \hfill -& \hfill 2x_2& \hfill =& \hfill 1\\ \hfill 3x_1& \hfill -& \hfill 6x_2& \hfill =& \hfill 3\end{array}$$
2. $$\begin{array}{ccccc}\hfill 3x_1& \hfill +& \hfill 2x_2& \hfill =& \hfill 7\\ \hfill -6x_1& \hfill -& \hfill 4x_2& \hfill =& \hfill 5\end{array}$$

First write the systems in matrix form and find $\left|A\right|$ :

Answer

Now compare the two equations in each system in turn:

Answer