Invertibility Proof for matrix

Question

Suppose that A is a square matrix that satisfies $A^n=0$ for some positive integer n. Show that $I-A$ is invertible and $(I-A)^{-1}=I+A+A^2+...+A^{n-1}$.

Not sure how to start the problem.

Answer 1

Hint 1. If the inverse $B$ exists, then it is defined uniquely by the property $$ (I-A)B=I $$ This problem gives you a candidate for $B$.

Hint 2. The polynomial $a^n-t^n$ factors as $$ a^n-t^n=(a-t)\left(a^{n-1}+a^{n-2}t+\dotsb +at^{n-2}+t^{n-1}\right) $$