endomorphisms problem, first year of college linear algebra

Question

Let f,g be 2 endomophisms on V(n=dimV>=2) such that

$$f\circ g+a f+a g=0 $$ Show that : a) $$f\circ g=g\circ f$$ b) $$rg(f)=rg(g)$$ c)for n and a given, give an example of f and g bijective endomorphisms.

Answer 1

The statement is false when $a=0$, since $f\circ g = 0$ does not imply $g\circ f =0$.

If $a\neq 0$, then the relation can be written as $(a^{-1} f + I) \circ (a^{-1} g + I) = I$, where $I$ is the identity. That does imply $(a^{-1} g + I) \circ (a^{-1} f + I) = I$, and from here conclude that $f\circ g = g \circ f$.