Let $A = [a_{ij}]$,where $a_{ij} \in \mathbb{Z} \cap [0, 4]$ and $1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2, 13)$ is $........$.

Which of the following determinant$(s)$ vanish(es)?

Let $I$ be an identity matrix of order $2 \times 2$ and $P = \begin{bmatrix} 2 & -1 \\ 5 & -3 \end{bmatrix}$. Then the value of $n \in N$ for which $P^n = 5I - 8P$ is equal to ..... .

If $A = \begin{bmatrix} \cos^2 \alpha & \cos \alpha \sin \alpha \\ \cos \alpha \sin \alpha & \sin^2 \alpha \end{bmatrix}$ and $B = \begin{bmatrix} \cos^2 \beta & \cos \beta \sin \beta \\ \cos \beta \sin \beta & \sin^2 \beta \end{bmatrix}$ are two matrices such that the product $AB$ is a null matrix,then $\alpha - \beta$ is:

If $\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$ is a skew-symmetric matrix and $b, c, f$ are non-zero real numbers,then $\frac{b}{c} = $

If $A = \begin{bmatrix} 3 & -2 \\ 4 & 2 \end{bmatrix}$,then $A^2 - 5A + 14I = 0$. Which of the following is equivalent to $A^2$?