Question

Solve the following system of equations using an inverse matrix. You must alsoindicate the inverse matrix, A-1, that was used to solve the system. You mayoptionally write the inverse matrix with a scalar coefficient.2x-3y = -55x - 4y = -2Al=y =

Answer 1

The two equations given are:

$\begin{gathered} 2x-3y=-5 \\ 5x-4y=-2 \end{gathered}$

The coefficient matrix A is:

$A=\begin{bmatrix}2 & -3 \\ 5 & -4\end{bmatrix}$

The variable matrix X is:

$X=\begin{bmatrix}x \\ y\end{bmatrix}$

and the constant matrix B is:

$B=\begin{bmatrix}-5 \\ -2\end{bmatrix}$

Then, AX = B looks like,

$\begin{gathered} AX=B \\ X=A^{-1}B \end{gathered}$

So, the variables "x" and "y" are found my multiplying the inverse of A by the matrix B.

Let's find the inverse matrix of A:

Given, a 2 x 2 matrix,

$A=\begin{bmatrix}a & b \\ c & d\end{bmatrix}$

The inverse of this matrix will be,

$A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$

Using the formula, we have:

$\begin{gathered} A^{-1}=\frac{1}{-8--15}\begin{bmatrix}-4 & 3 \\ -5 & 2\end{bmatrix} \\ =\frac{1}{7}\begin{bmatrix}-4 & 3 \\ -5 & 2\end{bmatrix} \\ =\begin{bmatrix}-\frac{4}{7} & \frac{3}{7} \\ -\frac{5}{7} & \frac{2}{7}\end{bmatrix} \end{gathered}$

Now, we can solve for the matrix X, shown below:

$X=\begin{bmatrix}-\frac{4}{7} & \frac{3}{7} \\ -\frac{5}{7} & \frac{2}{7}\end{bmatrix}\begin{bmatrix}-5 \\ -2\end{bmatrix}=\begin{bmatrix}(-\frac{4}{7})(-5)+(\frac{3}{7})(-2) \\ (-\frac{5}{7})(-5)+(\frac{2}{7})(-2)\end{bmatrix}=\begin{bmatrix}\frac{20}{7}-\frac{6}{7} \\ \frac{25}{7}-\frac{4}{7}\end{bmatrix}=\begin{bmatrix}\frac{14}{7} \\ \frac{21}{7}\end{bmatrix}=\begin{bmatrix}2 \\ 3\end{bmatrix}$

The solution matrix, X, is

$X=\begin{bmatrix}2 \\ 3\end{bmatrix}$

This, means the solution to the system of equations is:

$x=2,y=3$