Let $\alpha$ and $\beta$ be real numbers. Consider a $3 \times 3$ matrix $A$ such that $A ^2=3 A +\alpha I$. If $A ^4=21 A +\beta I$, then

Show that

$\left[ {\begin{array}{*{20}{l}}
  1&2&3 \\ 
  0&1&0 \\ 
  1&1&0 
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  { – 1}&1&0 \\ 
  0&{ – 1}&1 \\ 
  2&3&4 
\end{array}} \right]$ $ \ne \left[ {\begin{array}{*{20}{c}}
  { – 1}&1&0 \\ 
  0&{ – 1}&1 \\ 
  2&3&4 
\end{array}} \right]\left[ {\begin{array}{*{20}{l}}
  1&2&3 \\ 
  0&1&0 \\ 
  1&1&0 
\end{array}} \right]$

If $X = \left[ {\begin{array}{*{20}{c}}3&{ – 4}\\1&{ – 1}\end{array}} \right]$, then the value of ${X^n}$ is

If $A$ is an idempotent matrix, then $(I + A)^4$ is (where $I$ is identity matrix of order same as $A$ )

Let $\mathrm{p}$ be an odd prime number and $\mathrm{T}_{\mathrm{p}}$ be the following set of $2 \times 2$ matrices :

$T_p=\left\{A=\left[\begin{array}{ll}\mathrm{a} & \mathrm{b} \\ \mathrm{c} & \mathrm{a}\end{array}\right]: \mathrm{a}, \mathrm{b}, \mathrm{c} \in\{0,1, \ldots ., \mathrm{p}-1\}\right\}$

$1.$ The number of $A$ in $T_p$ such that $A$ is either symmetric or skew-symmetric or both, and $\operatorname{det}(\mathrm{A})$ divisible by $\mathrm{p}$ is

$(A)$ $(\mathrm{p}-1)^2$  $(B)$ $2(\mathrm{p}-1)$

$(C)$ $(\mathrm{p}-1)^2+1$  $(D)$ $2 \mathrm{p}-1$

$2.$ The number of $A$ in $T_p$ such that the trace of $A$ is not divisible by $p$ but det $(A)$ is divisible by $p$ is [Note: The trace of a matrix is the sum of its diagonal entries.]

$(A)$ $(\mathrm{p}-1)\left(\mathrm{p}^2-\mathrm{p}+1\right)$ $(B)$ $\mathrm{p}^3-(\mathrm{p}-1)^2$

$(C)$ $(\mathrm{p}-1)^2$ $(D)$ $(p-1)\left(p^2-2\right)$

$3.$ The number of $A$ in $T_p$ such that det $(A)$ is not divisible by $p$ is

$(A)$ $2 \mathrm{p}^2$ $(B)$ $p^3-5 p$ $(C)$ $p^3-3 p$ $(D)$ $p^3-p^2$

Give the answer question $1,2$ and $3.$