Let $S_1$ and $S_2$ be respectively the sets of all $a\in R-\left\{0\right\}$ for which the system of linear equations
$ax+2ay-3az=1$

$\left(2a+1\right) x+\left(2a+3\right) y+\left(a+1\right)z=2$

$\left(3a+5\right) x+\left(a+5\right) y+\left(a+2\right) z=3$

has unique solution and infinitely many solutions. Then

Let $\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{10}}& \frac{3}{\sqrt{10}}\\ \frac{-3}{\sqrt{10}}& \frac{1}{\sqrt{10}}\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{rr}1& -\mathrm{i}\\ 0& 1\end{array}\right]$, where $\mathrm{i}=\sqrt{-1}$. If $\mathrm{M}={\mathrm{A}}^{\mathrm{T}}\mathrm{BA}$, then the inverse of the matrix ${\mathrm{AM}}^{2023} {\mathrm{A}}^{\mathrm{T}}$ is

Let $A,B,C$ be $3\times 3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric.

Consider the statements

$\left(S1\right) A^{13} B^{26}-B^{26} A^{13}$ is symmetric

$\left(\mathrm{S}2\right) A^{26}{C}^{13}-C^{13} A^{26}$ is symmetric

Then,

Consider the following system of equations

$\alpha x+2y+z=1$

$2\alpha x+3y+z=1$

$3x+\alpha y+2z=\beta$

For some $\alpha ,\beta \in \mathrm{ℝ}$. Then which of the following is NOT correct.

The set of all values of $t\in \mathrm{ℝ}$, for which the matrix $\left[\begin{array}{ccc}{\mathrm{e}}^t& {\mathrm{e}}^{-t}\left(\sin t-2\cos t\right)& {\mathrm{e}}^{-t}\left(-2\sin t-\cos t\right)\\ {\mathrm{e}}^t& {\mathrm{e}}^{-t}\left(2\sin t+\cos t\right)& {\mathrm{e}}^{-t}\left(\sin t-2\cos t\right)\\ {\mathrm{e}}^t& {\mathrm{e}}^{-t}\cos t& {\mathrm{e}}^{-t}\sin t\end{array}\right]$ is invertible, is