If $P =\begin{bmatrix}5&3\\-1&-2\end{bmatrix}$ satisfies the equation $P^2-3P-7I=0$, where $I$ is an identity matrix of order 2, then $P^{-1}$ is:

$\frac{1}{7}\begin{bmatrix}2&3\\-1&-5\end{bmatrix}$

The correct answer is Option (1) → $\frac{1}{7}\begin{bmatrix}2&3\\-1&-5\end{bmatrix}$

$P =\begin{bmatrix}5&3\\-1&-2\end{bmatrix}$

Co-factor Matrix of $P=\begin{bmatrix}-2&1\\-3&5\end{bmatrix}$

$Adj\,P=\begin{bmatrix}-2&-3\\1&5\end{bmatrix}$

$|P|=5×-2-3×(-1)=-7$

$∴A^{-1}=\frac{1}{|P|}Adj\,P=\frac{1}{7}\begin{bmatrix}2&3\\-1&-5\end{bmatrix}$