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 a  $3\times 1$ column matrix of unknown variables and O is a  $3\times 1$ null matrix, has

$Given A^T=A, B^T=-B$ given system is homogenous system of equations
Determinant of coefficient matrix  is $\left|A^2{B}^2-B^2{A}^2\right|$

$\begin{array}{rcl}Consider {\left(A^2{B}^2-B^2{A}^2\right)}^T& =& {\left(A^2{B}^2{)}^T-(B^2{A}^2\right)}^T\\ & =& \left(B^2{A}^2\right)-\left(A^2{B}^2\right)\\ & =& -\left(A^2{B}^2-B^2{A}^2\right)\end{array}$

Therefore, $\left(A^2{B}^2-B^2{A}^2\right)$ is a skew symmetric matrix of odd order

Hence the determinant of the coefficient matrix is zero. 

Therefore, the system of equations have non trivial solutions, and it has infinitely many solutions