If a $2^{nd}$ order determinant with elements $0$ or $1$ is chosen from the set of all such determinants,find the probability that the determinant chosen is non-zero.

If $A$ and $B$ are two matrices such that $AB = B$ and $BA = A,$ then ${A^2} + {B^2} = $

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:
$(A) 2p^2$ $(B) p^3-5p$ $(C) p^3-3p$ $(D) p^3-p^2$

Let $A = \begin{bmatrix} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{bmatrix}$,where $a, c \in \mathbb{R}$. If $A^3 = A$ and the positive value of $a$ belongs to the interval $(n-1, n]$,where $n \in \mathbb{N}$,then $n$ is equal to $..........$.

Let $A = \begin{bmatrix} \cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta \end{bmatrix}$. If for some $\theta \in (0, \pi)$,$A^2 = A^T$,then the sum of the diagonal elements of the matrix $(A + I)^3 + (A - I)^3 - 6A$ is equal to . . . . . . .

For $A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 1 & 3 \\ 1 & 1 & 0 \end{bmatrix}$,if $A^3 - 2A^2 + kA - 4I_3 = 0$,then $k = $ . . . . . . .