For any square matrix $A$, $A{A^T}$ is a

Let $P =\left[\begin{array}{ccc}3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0\end{array}\right]$ where $\alpha \in R .$ Suppose $Q =\left[ q _{ ij }\right]$ is a matrix satisfying $PQ = kI _{3}$ for some non-zero $k \in R .$ If $q_{23}=-\frac{k}{8}$ and $|Q|=\frac{k^{2}}{2}$, then $\alpha^{2}+ k ^{2}$ is equal to………..

If $A = \left[ {\begin{array}{*{20}{c}}
1&1\\
0&1
\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}
{\frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}\\
{\frac{{ – 1}}{2}}&{\frac{{\sqrt 3 }}{2}}
\end{array}} \right]$ , then $(BB^TA)^5$ is equal to

Given $A$ and $C$ are involutary matrices and $B$ is a non-singular matrix, then $(AB^{-1}C)^{-1}$ is equal to –

Let $A$ be a non-zero periodic matrix with period $4$ and $A^{12} + B =I$, where $I$ is identity matrix and $B$ is any square matrix of same order as of $A$. Matrix product $AB$ is equal to