# Why does $A^2=I$ imply $nullity(A)=0$?

$A$ is a square matrix, why does $A^2=I$ imply $nullity(A)=0$?

• Thanks for this post. With little modification I just generated a nice problem for my lin algebra students... – imranfat May 14 '14 at 14:49

Suppose that $Ax = 0$ for some $x$. Then also $A^2 x = 0$, but $A^2 x = I x = x$, so $x = 0$.
If you know that invertible matrices have nullity $\{0\}$, then you could just observe that $A$ is invertible.
• Excellent observation. It is invertible because there exists a matrix $B$ such that $AB=BA=I$. In fact, $B=A$. – mathse May 14 '14 at 13:16