Given an $n \times n$ matrix $A$, if $Ax = x$ for all $x \in \Bbb R^n$, prove that $A$ is the identity matrix. How can I prove that this statement is true? I found this in an old textbook I was flipping through and was wondering how I could construct a proof for it.

Prove that if $Ax=x$ for all $x$ then $AI=I$ where $I$ is the identity matrix. 
