If $P$ is a $3 \times 3$ real matrix such that $P^{T} = aP + (a - 1)I$,where $a > 1$,then $..........$

If $B=\left[\begin{array}{ll}1 & 3 \\ 1 & \alpha\end{array}\right]$ is the adjoint of a matrix $A$ and $|A|=2$,then the value of $\alpha$ is

Consider the following statements:
Statement $I$: If $A$ is a non-singular matrix, then $A^{-1}$ exists.
Statement $II$: If $A$ and $B$ are symmetric matrices of the same order, then $(AB - BA)$ is a skew-symmetric matrix.
Choose the correct option.

Let $A = \begin{bmatrix} 2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix}$. If $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(2A)))| = (16)^n$,then $n$ is equal to

The inverse of $\begin{bmatrix} 2 & -3 \\ -4 & 2 \end{bmatrix}$ is

Let $A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} 6 & -13 \\ 5 & -10 \end{bmatrix}$ be two matrices. If the variables $x$ and $y$ satisfy the matrix equation $((A^{-1})^2 + B) \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$, then the ordered pair $(x, y) =$?