Let A and B be 3 × 3 real matrices such that A is a symmetric...

Let ${A}^{T} = A and {B}^{T} = - B$

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

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

$= {\left({B}^{2}\right)}^{T}{\left({A}^{2}\right)}^{T} - {\left({A}^{2}\right)}^{T}{\left({B}^{2}\right)}^{T}$

$= {B}^{2}{A}^{2} - {A}^{2}{B}^{2}$

${C}^{T} = - C$

$C$ is skew symmetric.

So det \left(C\right) = 0$

so system have infinite solutions.