Prove that the matrix $I_n - A$ is invertible Let $A$ be an $n \times n$ matrix with the property that $A^3 = O_n$, where $O_n$ denotes the $n \times n$ matrix which
has all the entries equal to $0$. Let $I_n$ be the $n \times n$ identity matrix. Prove that the matrix $I_n - A$
is invertible, and indicate how you would obtain its inverse.

Hint. For all $x\in\mathbb{R}$, such that $|x|<1$, one has: $$\frac{1}{1-x}=\sum_{k=0}^{+\infty}x^k.$$
Can you see how to adapt this to find the inverse of $I-A$, where $A$ is nilpotent?
