Use $tr(AB)=tr(BA)$ to show that the identity matrix $I_n$ is not contained in the span of the set $\{[A,B]|A,B\in M_{n\times n}(K)\}$. Note that it is not sufficient to show that $I_n\neq [A,B]$ for any pair $A,B$ of $n\times n$ matrices (8 pts).
I know how to prove the problem. But why it’s not sufficient?
I’m not sure if my following proof is correct. For any $A,B$, $tr([A,B])=tr(AB)-tr(BA)=0\neq n =tr(I_n)$. Thus $I_n\neq [A,B]$.
I think it’s sufficient to prove the problem.