Find a matrix A such that $\operatorname{rank}{A} = \operatorname{rank}{A^2} \neq \operatorname{rank}{A^3}$ Let $A$ be a complex square matrix of order 2 ($A \in M_{2,2}$).
Then, does there exist $A$ such that $\operatorname{rank}{A} = \operatorname{rank}{A^2} \neq \operatorname{rank}{A^3}$?
If that doesn't exist, how can I prove it?

The only way for a matrix to lose rank as you take powers, is through its Jordan blocks corresponding to zero eigenvalues.
