Let $A$ and $B$ be $3\times 3$ real matrices such that $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix. Then the system of linear equations $\left(A^2 B^2-B^2 A^2\right)X=O,$ where $X$ is a $3\times 1$ column matrix of unknown variables and $O$ is a $3\times 1$ null matrix, has

Consider the following system of equations: 

$x+2y-3z=a$

$2x+6y-11z=b$

$x-2y+7z=c$

where $a,b$ and $c$ are real constants. Then the system of equations :

Consider the system of linear equations

$-x+y+2z=0$

$3x-ay+5z=1$

$2x-2y-az=7$

Let $S_1$ be the set of all $a\in R$ for which the system is inconsistent and $S_2$ be the set of all $a\in R$ for which the system has infinitely many solutions. If $\mathrm{n}\left({\mathrm{S}}_1\right)$ and $\mathrm{n}\left({\mathrm{S}}_2\right)$ denote the number of elements in ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ respectively, then

If $x+ay+a^{2}z=0$,  $x+by+b^{2}z=0$, $x+cy+c^{2}z=0$ ,  $a\ne b\ne c$,  then given system of equations -