find the inverse of the matrix below.\n\\\\begin{bmatrix}-6&-12\\\\-9&-16\\end{bmatrix}\\\n\\\\begin{bmatrix}…

Explanation:

Step1: Calculate the determinant

For a 2x2 matrix $A=\begin{bmatrix}a&b\c&d\end{bmatrix}$, the determinant $\text{det}(A)=ad - bc$. Here $a=-6$, $b = - 12$, $c=-9$, $d=-16$. So $\text{det}(A)=(-6)\times(-16)-(-12)\times(-9)=96 - 108=-12$.

Step2: Swap elements and multiply by reciprocal of determinant

The inverse of a 2x2 matrix $A=\begin{bmatrix}a&b\c&d\end{bmatrix}$ is $A^{-1}=\frac{1}{\text{det}(A)}\begin{bmatrix}d&-b\-c&a\end{bmatrix}$. Substituting values, we get $A^{-1}=\frac{1}{-12}\begin{bmatrix}-16&12\9&-6\end{bmatrix}=\begin{bmatrix}\frac{-16}{-12}&\frac{12}{-12}\\frac{9}{-12}&\frac{-6}{-12}\end{bmatrix}=\begin{bmatrix}\frac{4}{3}& - 1\-\frac{3}{4}&\frac{1}{2}\end{bmatrix}\approx\begin{bmatrix}1.33&-1\ - 0.75&0.50\end{bmatrix}$

Answer:

$\begin{bmatrix}1.33&-1\-0.75&0.50\end{bmatrix}$