Rank is less than $\frac n2$

Question

Let $A \in \mathbb{R}^{n \times n}$ such that $A^2 =0$. Prove that $\mbox{rank}(A) \leq \frac n2$.

With Cayley-Hamilton, the characteristic polynomial is $\chi_A=X^2$. I also know $\dim A = \dim(Im(A)) + \dim(Kernel(A))$ so $n = \dim(Im(A)) + \dim(Kernel(A))$.

Answer 1

Hint:

$Im(A) \subset ker A$ and see what will happen!