Prove $\text{rank}(A^2) — \text{rank}(A^3) \leq \text{rank}(A) — \text{rank}(A^2)$

Question

Let $A$ be an $n \times n$ matrix over a field $K$. Prove that $\text{rank}(A^2) — \text{rank}(A^3) \leq \text{rank}(A) — \text{rank}(A^2)$

In particular - I am not aware of any relationship that must hold in general between $\text{rank}(A)$ and $\text{rank}(A^2)$.

Answer 1

Hint: Consider epimorphism $\operatorname{im}A\to \operatorname{im}A^2/\operatorname{im}A^3$ given by $x\mapsto Ax + \operatorname{im}A^3$ which factors through $\operatorname{im}A/\operatorname{im}A^2$.