Find the inverse matrix of $\begin{bmatrix} 4 & 7 \\ 1 & 2 \end{bmatrix}$.

If $B = \begin{bmatrix} 3 & \alpha & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3 \end{bmatrix}$ is the adjoint of a $3 \times 3$ matrix $A$ and $|A| = 4$,then $\alpha$ is equal to:

Let $A$ be a $3 \times 3$ matrix such that $A+A^{T}=O$. If $A\begin{bmatrix}1\\ -1\\ 0\end{bmatrix}=\begin{bmatrix}3\\ 3\\ 2\end{bmatrix}$,$A^{2}\begin{bmatrix}1\\ -1\\ 0\end{bmatrix}=\begin{bmatrix}-3\\ 19\\ -24\end{bmatrix}$ and $\det(\text{adj}(2\text{adj}(A+I))) = (2)^\alpha \cdot(3)^\beta \cdot(11)^\gamma$,then $\alpha+\beta+\gamma$ is equal to . . . . . . .

If $\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} \cdot A \cdot \begin{bmatrix} -3 & 2 \\ 5 & -3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$,then $A =$

If $A^2-A+I=0$,then the inverse of the matrix $A$ is

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