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 \(\left(A^2 B^2-B^2 A^2\right) X=0\). where \(X\) is \(3 \times 1\) column matrix of unknown variables and O is a \(3 \times 1\) null matrix, has

Let \(P=A^2 B^2-B^2 A^2\)
\(\therefore \mathrm{P}^{\mathrm{T}} =\left(\mathrm{A}^2 \mathrm{~B}^2-\mathrm{B}^2 \mathrm{~A}^2\right)^{\mathrm{T}} \)
\( =\left(\mathrm{A}^2 \mathrm{~B}^2\right)^{\mathrm{T}}-\left(\mathrm{B}^2 \mathrm{~A}^2\right)^{\mathrm{T}} \)
\( =\left(\mathrm{B}^2\right)^{\mathrm{T}}\left(\mathrm{~A}^2\right)^{\mathrm{T}}-\left(\mathrm{A}^2\right)^{\mathrm{T}}\left(\mathrm{~B}^2\right)^{\mathrm{T}} \)
\( =\mathrm{B}^2 \mathrm{~A}^2-\mathrm{A}^2 \mathrm{~B}^2 \quad \cdots\left[\because \mathrm{~A}^{\mathrm{T}}=\mathrm{A} \text { and } \mathrm{B}^{\mathrm{T}}=-\mathrm{B}\right] \)
\( =-\left(\mathrm{A}^2 \mathrm{~B}^2-\mathrm{B}^2 \mathrm{~A}^2\right) \)
\( =-\mathrm{P}\)
\(\therefore P\) is a skew - symmetric matrix.
\(\therefore \operatorname{det}(\mathrm{P})=0\)
\(\therefore \) The given system of equations has infinitely many solutions.