Nilpotent matrices and inverses

Question

Can somebody give me a hint for showing that:

The matrix $A+I$ is invertible if there is an integer $k\gt 0$ so that $A^k=0$.

Answer 1

Hint

$$(1+x)\sum_{p=0}^\infty(-1)^p x^p=1$$

Answer 2

Consider the formula for summing the finite geometric series $$I+(-A)+(-A)^2+\cdots+(-A)^{k-1}$$ (since everything commutes, the basic algebraic manipulations involved here are valid). The formula requires dividing by $I-(-A)=I+A$, which is precisely what you are trying to justify here; as a consequence this does not give you a proof, just an idea. However the proof of the geometric series formula is by multiplying out the denominator, and that can be validly done here.