Let $A$ be a $2 \times 2$ matrix of the form $A = \begin{bmatrix} a & b \\ 1 & 1 \end{bmatrix}$,where $a, b$ are integers and $-50 \leq b \leq 50$. The number of such matrices $A$ such that $A^{-1}$,the inverse of $A$,exists and $A^{-1}$ contains only integer entries is

If $A$ and $B$ are square matrices of order $3$ such that $|A|=2$ and $|B|=4$,then $|A(\operatorname{adj} B)| = \dots$

If $|A| = 2$,where $A$ is a square matrix of order $4$,then the value of $|Adj(Adj(2A))|$ is (where $Adj(A)$ denotes the adjoint of matrix $A$):

If $A$ is a $2 \times 2$ order non-singular matrix,then the determinant of $A^{-1}$ is . . . . . . .

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 . . . . . . .

Let $A$ be any $3 \times 3$ invertible matrix. Then which one of the following is not always true?