Let $\mathrm{A}$ and $\mathrm{B}$ be $3 \times 3$ real matrices such that $\mathrm{A}$ is symmetric matrix and $B$ is skew-symmetric matrix. Then the system of linear equations $\left(\mathrm{A}^{2} \mathrm{~B}^{2}-\mathrm{B}^{2} \mathrm{~A}^{2}\right) \mathrm{X}=\mathrm{O}$, where $\mathrm{X}$ is a $3 \times 1$ column matrix of unknown variables and $\mathrm{O}$ is a $3 \times 1$ null matrix, has:
Correct Option: , 3
Let $\mathrm{A}^{\mathrm{T}}=\mathrm{A}$ and $\mathrm{B}^{\mathrm{T}}=-\mathrm{B}$
$\mathrm{C}=\mathrm{A}^{2} \mathrm{~B}^{2}-\mathrm{B}^{2} \mathrm{~A}^{2}$
$\mathrm{C}^{\mathrm{T}}=\left(\mathrm{A}^{2} \mathrm{~B}^{2}\right)^{\mathrm{T}}-\left(\mathrm{B}^{2} \mathrm{~A}^{2}\right)^{\mathrm{T}}$
$=\left(B^{2}\right)^{T}\left(A^{2}\right)^{T}-\left(A^{2}\right)^{T}\left(B^{2}\right)^{T}$
$=B^{2} A^{2}-A^{2} B^{2}$
$C^{T}=-C$
$\mathrm{C}$ is skew symmetric.
So $\operatorname{det}(C)=0$
so system have infinite solutions.
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