If $A=\begin{bmatrix} 1 & 0 & -2 \\ -2 & -1 & 2 \\ 3 & 4 & 1 \end{bmatrix}$,then $A^{-1}=$

If $P = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix}$ is the adjoint of a $3 \times 3$ matrix $A$ and $|A| = 4$,then the value of $\alpha$ is:

If $A$ is a $3 \times 3$ non-singular matrix such that $AA' = A'A$ and $B = A^{-1}A'$,then $BB'$ equals:

If $A, B$ and $(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1}))$ are non-singular matrices of the same order,then the inverse of $A(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1}))^{-1}B$ is equal to

Let $P = \begin{bmatrix} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{bmatrix}$ where $\alpha \in R$. Suppose $Q = [q_{ij}]$ 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 $A = \begin{bmatrix} 1 & 2 \\ -1 & 4 \end{bmatrix}$ and $A^{-1} = \alpha I + \beta A$,where $\alpha, \beta \in \mathbb{R}$ and $I$ is the identity matrix of order $2$,then $4(\alpha + \beta) = $