Q:

If A is a matrix satisfying $A^3 = A^2$ and the determinant of $A$ cannot equal $0$, must $A=I$.

$I$ is an identity matrix. 
How would one go about conducting a proof for something like this? 
I know that this doesn't work when $A$ is $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$
, are there any other cases where this statement is true and $A$ does not equal $I$? Any hints are appreciated!

A:

If the determinant of A is not 0, then you can multiply both sides of the equation by $(A^{-1})^2 $ to get $A=I$

