Use inverse matrices to find the solution of the system of equations.

Question

Use inverse matrices to find the solution of the system of equations. ​  ​

Quizplus · Accepted Answer

We can rewrite the system of equations as the matrix equation:

$\begin{bmatrix} 2 & 3 \ 5 & 4 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 9 \ 15 \end{bmatrix}$

To solve for $x$ and $y$, we can multiply both sides by the inverse of the coefficient matrix.

$\begin{bmatrix} 2 & 3 \ 5 & 4 \end{bmatrix}^{-1} \begin{bmatrix} 2 & 3 \ 5 & 4 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 2 & 3 \ 5 & 4 \end{bmatrix}^{-1} \begin{bmatrix} 9 \ 15 \end{bmatrix}$

Simplifying the left-hand side and multiplying the inverse matrix on the right-hand side, we get:

$\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} -3 & 3/2 \ 5/2 & -1 \end{bmatrix} \begin{bmatrix} 9 \ 15 \end{bmatrix}$

Simplifying the right-hand side, we get:

$\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} -18 \ 22 \end{bmatrix}$

So our solution is $x = -18$ and $y = 22$. Thus, the correct choice is (C).

Use Inverse Matrices to Find the Solution of the System