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)=(1)/(ad-bc)\begin{bmatrix}d & -b \\ -c & a\end{bmatrix}`

Using the formula, we have:

`\begin{gathered} A^(-1)=(1)/(-8--15)\begin{bmatrix}-4 & 3 \\ -5 & 2\end{bmatrix} \\ =(1)/(7)\begin{bmatrix}-4 & 3 \\ -5 & 2\end{bmatrix} \\ =\begin{bmatrix}-(4)/(7) & (3)/(7) \\ -(5)/(7) & (2)/(7)\end{bmatrix} \end{gathered}`

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

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