If $A = f(x) = \begin{bmatrix} \cos x & \sin x & 0 \\ -\sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$,then $A^{-1}$ is equal to:

Show that the matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ satisfies the equation $A^{2} - 4A + I = O$,where $I$ is the $2 \times 2$ identity matrix and $O$ is the $2 \times 2$ zero matrix. Using this equation,find $A^{-1}$.

If $A$ is a $3 \times 3$ matrix such that $|5 \cdot \text{adj } A| = 5$,then $|A|$ is equal to

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

If $A$ is a square matrix such that $A(\operatorname{adj} A) = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{bmatrix}$, then $\operatorname{det}(\operatorname{adj} A)$ is equal to

If $A = \begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{bmatrix}$,then $|A| |adj A|$ is equal to