Answer
$\begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}$
Work Step by Step
Let $P$ be the given matrix.
First we check if the matrix $P$ has an inverse:
$det P=\begin{vmatrix} 3 & -1 \\ -5 & 2 \end{vmatrix}=3(2)-(-5)(-1)=1$
Because $det P\not=0$, the matrixx $P$ has an inverse.
We will apply the formula for the inverse of $ 2\times 2$ matrix to obtain:
$P^{-1}=\dfrac{1}{detP} \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}$
$=\dfrac{1}{1}\begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}$
$=\begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}$
