If $A^2 = I$, then $\operatorname{rank} (A+I) + \operatorname{rank} (A-I) =n$

Question

If $A^2 = I$ where $A$ is an $n\times n$ matrix, then why is $\operatorname{rank} (A+I) + \operatorname{rank} (A-I) =n$.

I am not sure why this seems to be true.