Let $M$ and $N$ be two invertible square matrices over $\mathbb{R}$ of order $2$ such that $N$ is diagonal. Then $M N M^{-1}$ is diagonal . . . . . .

Consider the matrix $f(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Given below are two statements:
Statement $I$: $f(-x)$ is the inverse of the matrix $f(x)$.
Statement $II$: $f(x) f(y) = f(x+y)$.
In the light of the above statements,choose the correct answer from the options given below:

Let $\omega$ be a complex number such that $2\omega + 1 = z$ where $z = \sqrt{-3}$. If $\left| \begin{array}{ccc} 1 & 1 & 1 \\ 1 & -\omega^2 - 1 & \omega^2 \\ 1 & \omega^2 & \omega^7 \end{array} \right| = 3k$,then $k$ is equal to:

If $\begin{vmatrix} x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2 \end{vmatrix} = \frac{9}{8}(103x+81)$,then $\lambda$ and $\frac{\lambda}{3}$ are the roots of the equation:

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 $A$ and $B$ be two $3 \times 3$ real matrices such that $(A^{2}-B^{2})$ is an invertible matrix. If $A^{5}=B^{5}$ and $A^{3} B^{2}=A^{2} B^{3}$,then the value of the determinant of the matrix $A^{3}+B^{3}$ is equal to: