# Let A and B be $3 \times 3$ real matrices such that A is symmetric matrix and B is 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 : Option: 1 infinitely many solutions Option: 2 no solution Option: 3 a unique solution Option: 4 exactly two solutions

$\\ \text {Let } A^{T}=A \text { and } B^{T}=-B \\\\ C=A^{2} B^{2}-B^{2} A^{2} \\ \\C^{T}=\left(A^{2} B^{2}\right)^{T}-\left(B^{2} A^{2}\right)^{T} \\ \\C^{T}=\left(B^{2}\right)^{T}\left(A^{2}\right)^{T}-\left(A^{2}\right)^{T}\left(B^{2}\right)^{T}$

$\\ \\C^{T}= B^{2} A^{2}-A^{2} B^{2} \\\\ C^{T}=-C$

C is skew symmetric.

So $\operatorname{det}(C)=0$

so system have infinite solutions.

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