Let $A$ and $B$ be real matrices of the form $\begin{bmatrix} \alpha & 0 \\ 0 & \beta \end{bmatrix}$ and $\begin{bmatrix} 0 & \gamma \\ \delta & 0 \end{bmatrix}$,respectively.
Statement $1$: $AB - BA$ is always an invertible matrix.
Statement $2$: $AB - BA$ is never an identity matrix.

If the matrix $A = \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix}$ is expressed as the sum of a symmetric matrix $B$ and a skew-symmetric matrix $C$, then which of the following relations is correct?

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 $B=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$ and $A$ be a $2 \times 2$ matrix satisfying $\left(A^T\right)^{-1}=A$. If $X=A B A^T$, then $A^T X^{2021} A=$

Matrix $A$ satisfies $A^2 = 2A - I$,where $I$ is the identity matrix. Then for $n \ge 2$,$A^n$ is equal to $(n \in N)$:

If the function $f:[a, b] \rightarrow \left[-\frac{\sqrt{3}}{4}, \frac{1}{2}\right]$ defined by $f(x) = \left| \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1+\sin x & 1 \\ 1+\cos x & 1 & 1 \end{array} \right|$ is one-one and onto, then: