I'm clearly lacking insight on the following problem...
So let $C$ be a $2\times2$ matrix with entries in some field. I want to know when I can find $A, B$ (also $2\times2$) such that $C=AB-BA$. In particular, you should be able to find them iff ${\rm Tr}(C)=0$ right? I'm having trouble with the "given ${\rm Tr}(C)=0$, prove that $A, B$" exist direction.
Methinks the problem is intended to be done with minimal background...i.e. knowing what a determinant is is less than kosher (nor should I know that $AB-BA$ is $[A,B]$).