If $A = \begin{bmatrix} 7 & 3 \\ 5 & -7 \end{bmatrix}$ be such that $A^{-1} = kA$, then $k$ equals

The inverse of the matrix $\begin{bmatrix} 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6 \end{bmatrix}$ is

If $[2\ \ x\ \ 3]\begin{bmatrix}-1 & 0 & -1\\ -1 & 1 & 0\\ 0 & 1 & 1\end{bmatrix}\begin{bmatrix}x\\0\\-1\end{bmatrix}=[-4]$, then the value(/s) of $x$ is

If a matrix P is both symmetric and skew-symmetric, then

If $a$, $b$ and $c$ are distinct prime numbers then the value of $\begin{vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{vmatrix}$ is equal to

If $A=\begin{bmatrix}a & 0\\0 & 0\end{bmatrix}$ and $B=\begin{bmatrix}0 & 0\\0 & b\end{bmatrix}$ then the matrix $A^3B^3$ is:

If A and B are square matrices of order 3 such that $|A| = 2$, $|B| = 3$, and $|2A \cdot \text{adj}(3(\text{adj}B))| = 2^\alpha \cdot 3^\beta$, then value of $\alpha + \beta$ is: