[Math] If $\operatorname{rank}(A)$ = $\operatorname{rank}(A^2)$, show that nullspace of $A$ = nullspace of $A^2$

Question

Let $A$ be a square matrix.

If $\operatorname{rank}(A)$ = $\operatorname{rank}(A^2)$

Prove that nullspace of $A$ = nullspace of $A^2$

The first thing I notice is that this $\implies$ $\operatorname{nullity}(A)=\operatorname{nullity}(A^2)$

Then I am kinda stuck, any hints?

Answer 1

You should show that the nullspace of $A$ is contained in the nullspace of $A^2$. By your observation, the dimension of the two nullspaces must necessarily be the same, so $\text{nullspace} A \subseteq \text{nullspace}A^2$ necessarily gives that they are equal.