Answer
$\begin{bmatrix}2&-1\\-\frac{5}{2}&\frac{3}{2}\end{bmatrix}$
Work Step by Step
We are given the matrix:
$A=\begin{bmatrix} 3&2\\5&4\end{bmatrix}$
Compute $det A$:
$detA=\begin{vmatrix}3&2\\5&4\end{vmatrix}=3(4)-5(2)=2$
Because $detA\not=0$, the inverse of the matrix exists. We determine it using the formula:
$A^{-1}=\dfrac{1}{det A}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$,
where $detA=ad-bc$ and $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$
$A^{-1}=\dfrac{1}{2}\begin{bmatrix}4&-2\\-5&3\end{bmatrix}=\begin{bmatrix}2&-1\\-\frac{5}{2}&\frac{3}{2}\end{bmatrix}$
