Let $B=\begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix}$ and $A$ be a $2 \times 2$ matrix such that $AB^{-1}=A^{-1}$. If $BCB^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$,then $2\beta-\alpha$ is equal to:

Let $A = \begin{bmatrix} 2 & -1 \\ 0 & 2 \end{bmatrix}$. If $B = I - {}^{3}C_{1}(\operatorname{adj} A) + {}^{3}C_{2}(\operatorname{adj} A)^{2} - {}^{3}C_{3}(\operatorname{adj} A)^{3}$,then the sum of all elements of the matrix $B$ is

The total number of matrices $A = \begin{bmatrix} 0 & 2x & 2x \\ 2y & y & -y \\ 1 & -1 & 1 \end{bmatrix}$ where $x, y \in \mathbb{R}$ and $x \neq y$,for which $A^T A = 3I_3$ is:

If $A$ and $B$ are square matrices of the same order such that $AB = A$ and $BA = B$,then $(A + I)^5$ is equal to (where $I$ is the identity matrix).

If $A$ and $B$ are two invertible square matrices of the same order such that $(A + B)(A - B) = A^2 - B^2$,then $(A^2BA^{-1}B^{-1})^3$ is equal to-

If $A = \begin{bmatrix} p & q & r \\ r & p & q \\ q & r & p \end{bmatrix}$ and $A A^T = I$,then $p^3 + q^3 + r^3 =$ . . . . . .