Let $F$ be a linear operator such that $F^2 - F + I = 0$, show that $F$ is invertible and $F^{-1} = I - F$ I didn't understand this exercise. I tried working with
$$F^2 - F + I = 0\implies (F-I)(F) + I =0$$
but I really don't understand how to prove $F$ is invertible neither find the inverse. Any hints? Thank you so much!

Hint 1: can $0$ be an eigenvalue of $F$?
Hint 2: can you multiply $F^2-F+I=0$ by $F^{-1}$ once you know it exists?
