I have the next problem:

Let $A$ be a real square matrix such that $A ^ 2 = 2\mathbb{I}$. Prove that $A +\mathbb{I}$ is an invertible matrix and find its inverse. 

I tried with the answers given here:[What is inverse of $I+A$?](https://math.stackexchange.com/questions/298616/what-is-inverse-of-ia)

Any hints?