TITLE: Why does $A^2=I$ imply $nullity(A)=0$?
QUESTION [5 upvotes]: $A$ is a square matrix, why does $A^2=I$ imply $nullity(A)=0$?

This is the key step in the solution, which I can't get it. Please help

REPLY [14 votes]: Suppose that $Ax = 0$ for some $x$. Then also $A^2 x = 0$, but $A^2 x = I x = x$, so $x = 0$.

REPLY [7 votes]: If you know that invertible matrices have nullity $\{0\}$, then you could just observe that $A$ is invertible.
If you don't know this, then fuglede's excellent elemenary answer suffices.