Answer
$\begin{bmatrix} 2 & 1 \\ -\frac{3}{2} & -\frac{1}{2} \end{bmatrix}$
Work Step by Step
Let us suppose that $P$ is the given matrix.
First we check if the inverse exists:
$det P=\begin{vmatrix} -1 & -2 \\ 3 & 4 \end{vmatrix}=4(-1)-3(-2)=2$
Since $det P\not=0$, the inverse exists.
$P^{-1}=\dfrac{1}{2} \begin{bmatrix} 4 & 2 \\ -3 & -1 \end{bmatrix}$
$=\begin{bmatrix} 2 & 1 \\ -\frac{3}{2} & -\frac{1}{2} \end{bmatrix}$
