Answer
$\begin{bmatrix} -\frac{1}{5} & -\frac{2}{5} \\ \frac{2}{5} & -\frac{1}{5} \end{bmatrix}$
Work Step by Step
Let us suppose that $P$ is the given matrix.
First we check if the inverse exists:
$det P=\begin{vmatrix} -1 & 2 \\ -2 & -1 \end{vmatrix}=(-1)(-1)-2(-2)=5$
Since $det P\not=0$ the inverse of matrix $P$ exists.
We will apply the formula for the inverse of $ 2\times 2$ matrix to obtain:
$P^{-1}=\dfrac{1}{det P} \begin{bmatrix} -1 & -2 \\ 2 & -1 \end{bmatrix}$
$=\dfrac{1}{5} \begin{bmatrix} -1 & -2 \\ 2 & -1 \end{bmatrix}$
$=\begin{bmatrix} -\frac{1}{5} & -\frac{2}{5} \\ \frac{2}{5} & -\frac{1}{5} \end{bmatrix}$
