I wanted to prove/disprove whether for every matrix $A\in \operatorname{Mat}_{n \times n}$
$\operatorname{rank} (A^2) = \operatorname{rank}(A)$ for any matrix $A\in \operatorname{Mat}_{n \times n}$
I know that for two matrices $A, B$ (it actually doesn't matters what is the size of $B$ as long as $AB$ is defined)
$$\operatorname{rank}(AB) \leq \min \{ \operatorname{rank}(A), \operatorname{rank}(B)\}$$
Therefore, $$\operatorname{rank}(A^2) \leq \operatorname{rank} (A)$$
In order to prove that:
$$\operatorname{rank}(A^2) = \operatorname{rank}(A)$$
you'd have to prove that: $$\operatorname{rank}(A^2) \geq \operatorname{rank}(A)$$ which I don't know how to prove, or finding a different way to prove directly both directions (meaning; one direction is $\leq$ and the other is $\geq$)