Q:

proving two linear maps are isomorphisms

I'm trying to prove that given two endomorphisms $f$ and $g$, and $f \circ g = g \circ f = -3 id$ f and g are isomorphisms. I have a feeling this should be easy but simultaneously I don't even know where to start. Any help would be greatly appreciated!

A:

From what is given, $-\frac13 g$ is a two-sided inverse of $f$, hence $f$ is an isomorphism. The same argument applies for $g$.
(Note that this proof is only valid if we are allowed to divide by $3$)

