A problem regarding eigenvalues symmetric about origin

Question

Show that eigenvalues of $A, B$ and $AB$ are symmetric about origin. I am doing this problem but I cann't think eigenvalues of $A$ if order of $A$ is odd.In $3*3$ matrices say these are $x,y,z$.From the result we take $y=-x$ but what about the 3rd eigenvalue?To make it symmetric we cann't take it $0$ as $A$ is nonsingular.Same problem occurs for $5*5, 7*7, 9*9......$.How can I overcome this confusion or problem?

Answer 1

You are right, it does not work for odd dimensions. Note that the matrix $AB$ is skew-symmetric as $(AB)^T=B^TA^T=BA=-AB$. Then it must be singular. It contradicts to the assumption that $A$ and $B$ are non-singular.