If $A$ and $B$ are two non-zero $n\times n$ matrices such that $A^2+B=A^{2}B,$ then

Let $\alpha$ be a root of the equation $\left(a-c\right)x^2+\left(b-a\right)x+\left(c-b\right)=0$ where $a, b, c$ are distinct real numbers such that the matrix $\left[\begin{array}{ccc}{\alpha }^2& \alpha & 1\\ 1& 1& 1\\ a& b& c\end{array}\right]$ is singular. Then the value of  $\frac{{\left(a-c\right)}^2}{\left(b-a\right)\left(c-b\right)}+\frac{{\displaystyle {\left(b-a\right)}^2}}{\left(a-c\right){\displaystyle \left(c-b\right)}}+\frac{{\displaystyle {\left(c-b\right)}^2}}{\left(a-c\right){\displaystyle \left(b-a\right)}}$ is

If the system of equations $x+2y+3z=3$, $4x+3y-4z=4$ and $8x+4y-\lambda z=9+\mu$ has infinitely many solutions, then the ordered pair $\left(\lambda ,\mu \right)$ is equal to

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.