If $A$ is a singular matrix of order $n$,then $A(adj\,A)$ is

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 $A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix}$ and $A \text{ adj } A = \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$,then $k$ is equal to :-

If for the matrix $A$,${A^3} = I$,then ${A^{-1}} = $

If for a matrix $A$, $|A|=6$ and $\text{adj } A = \begin{bmatrix} 1 & -2 & 4 \\ 4 & 1 & 1 \\ -1 & k & 0 \end{bmatrix}$, then $k$ is equal to

If $A$ is a non-singular matrix such that $A \cdot A^T = A^T \cdot A$ and $B = A^{-1} \cdot A^T$,then