IMC 1994 Problem 4

Let $\alpha\in\mathbb R\backslash \{ 0 \}$ and suppose that $F$ and $G$ are linear maps (operators) from $\mathbb R^n$ into $\mathbb R^n$ satisfying $F\circ G - G\circ F=\alpha F$ .

a) Show that for all $k\in\mathbb N$ one has $F^k\circ G-G\circ F^k=\alpha kF^k$ .

b) Show that there exists $k\geq 1$ such that $F^k=0$ .