If $AB + BA = 0$, what is $\operatorname{tr}(B)$? If $A$ is a regular square matrix with the dimension $n$ and $B$ is a square matrix with the dimension of $n$ and $AB + BA = 0$, find the trace of $B$.
From here, I can get this: 
$AB = -BA$
$ABA^{-1} = -B$
But I just don't see how I should proceed, or if I am even on the right track. Any help with the solution is appreciated..

We have 
$$
\operatorname{tr}(B)=\operatorname{tr} (ABA^{-1})=\operatorname{tr}(-B)=-
\operatorname{tr}(B).
$$
This implies $2\cdot \operatorname{tr}(B)=0$
For fields of characteristic not $2$ this means that $\operatorname{tr}(B)=0$, but for fields of characteristic $2$ this is not true.
