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!
2026-03-26 17:29:45.1774546185
proving two linear maps are isomorphisms
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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 $3$)