If $\det(AB-BA) = 0$ and $A^2 + B^2 = 0_n$, then show that $n$ is even.

Question

Let $A,B ∈ M_n(\Bbb R)$. Suppose that $A^2 + B^2 = 0_n$ and $\det( AB - BA ) =0$. Show that $n$ is even.

Answer 1

A simple counterexample is $A=B=0$ for any odd value of $n$.

On the other hand, if either $A$ or $B$ is invertible, then there is a very simple proof.

$A^2=-B^2$ therefore $|A^2|=|-B^2|$ and so $|A|^2=(-1)^n|B|^2$.

Then $(-1)^n$ is positive and so $n$ is even.