Let $A = \begin{bmatrix} 0 & 2q & r \\ p & q & -r \\ p & -q & r \end{bmatrix}$. If $AA^T = I_3$,then $|p|$ is

Let $A$ be a $3 \times 3$ matrix and $B$ be its adjoint matrix. If $|B|=64,$ then $|A|$ is equal to

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 $2b + 5c + 10d =$

Find the inverse of the matrix,if it exists: $\left[\begin{array}{cc}6 & -3 \\ -2 & 1\end{array}\right]$

If $A$ is a square matrix of order $3 \times 3$ and $\operatorname{det}(A) = 3$,then the value of $\operatorname{det}(3A^{-1})$ is:

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: