I have the following question:
Suppose A is a linear transformation. Show that if $A^2 - A + I = O$, then A is invertible.
To show $A$ is invertible, I can show if $Ax = 0$ then $x = 0$.
But, if $Ax = 0$, then $(Ax)^2 + (Ax) + 1 = 0 + 0 + 1 \neq 0$ so the equation in the question is not satisfied. What gives?