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:

Let $p$ be an odd prime number and $T_{p}$ be the set of $2 \times 2$ matrices defined as:
$T_p = \left\{ A = \begin{bmatrix} a & b \\ c & a \end{bmatrix} : a, b, c \in \{0, 1, \ldots, p-1\} \right\}$
$1.$ The number of matrices $A \in T_p$ such that $A$ is either symmetric or skew-symmetric or both,and $\det(A)$ is divisible by $p$ is:
$(A) (p-1)^2$ $(B) 2(p-1)$ $(C) (p-1)^2+1$ $(D) 2p-1$
$2.$ The number of matrices $A \in T_p$ such that the trace of $A$ is not divisible by $p$ but $\det(A)$ is divisible by $p$ is:
$(A) (p-1)(p^2-p+1)$ $(B) p^3-(p-1)^2$ $(C) (p-1)^2$ $(D) (p-1)(p^2-2)$
$3.$ The number of matrices $A \in T_p$ such that $\det(A)$ is not divisible by $p$ is:
$(A) 2p^2$ $(B) p^3-5p$ $(C) p^3-3p$ $(D) p^3-p^2$

If the matrix $A=\begin{bmatrix} 0 & 2 \\ K & -1 \end{bmatrix}$ satisfies $A(A^{3}+3I)=2I$,then the value of $K$ is:

If $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix}$,prove that $A^{3} - 6A^{2} + 7A + 2I = 0$.

If $A$ is a matrix such that $A^2 + A + 2I = O$,then which of the following is $INCORRECT$?

The value of $(1+\Delta)(1-\nabla)$ is