Question 23
Suppose $A, B \in \mathcal{M}_n(\mathbb{C})$ are such that $A + B = AB + BA$. Prove that:
a) if $n$ is odd, then $\det(AB - BA) = 0$;
b) if $\operatorname{tr}(A) \ne \operatorname{tr}(B)$, then $\det(AB - BA) = 0$.
a) if $n$ is odd, then $\det(AB - BA) = 0$;
b) if $\operatorname{tr}(A) \ne \operatorname{tr}(B)$, then $\det(AB - BA) = 0$.