Let $A = \begin{bmatrix} 1 & 2 \\ -2 & 1 \end{bmatrix}$ and $B^{-1} = \begin{bmatrix} 1 & 1 \\ 0 & 2 \end{bmatrix}$. If $(A B^{-1})^{-1} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$,then $2b + 5c + 10d =$

Let $P=\begin{bmatrix} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{bmatrix}$,where $\alpha \in \mathbb{R}$. Suppose $Q=[q_{ij}]$ is a matrix such that $PQ=kI$,where $k \in \mathbb{R}, k \neq 0$ and $I$ is the identity matrix of order $3$. If $q_{23}=-\frac{k}{8}$ and $\det(Q)=\frac{k^2}{2}$,then:

If the inverse of $\begin{bmatrix} 1 & 2 & x \\ 4 & -1 & 7 \\ 2 & 4 & -6 \end{bmatrix}$ does not exist,then $x=$

The adjoint of the matrix $A = \begin{bmatrix} 2 & -3 \\ 3 & 5 \end{bmatrix}$ is

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

Let $A$ be a $3 \times 3$ real matrix. If $\det(2 \operatorname{Adj}(2 \operatorname{Adj}(\operatorname{Adj}(2 A))))=2^{41}$,then the value of $\det(A^{2})$ is equal to ..... .