If $A$ is an $n \times n$ matrix and $ A^2 = 0$, then $\text{rank}(A)\le n/2$. Was revising for a Linear Algebra when I came across this question.

"Given a Matrix $A$ over $\mathbb{R}^{n \times n}$, and $A^2 = 0$,
  show that rank($A$) $\leq n/2$"

My attempt:
I was thinking of the theorem from Cayley-Hamilton but the matrix is a $n \times n$ matrix hence that's probably invalid.
Is there perhaps a significance of $A^2 = 0$? apart from a special case of the nilpotent matrix?

Hint: Let $k = \dim\ker A$. What can you say about $\dim\ker A^2$? Now use $\dim\ker A^2 + \mathop{\rm rank} A^2 = n$.
