What is $\mathrm{rank}(A)$ if it is known that $A^4 = 0$? 
Given $A \in M_{5 \times 5} (\mathbb F)$, what are the options for $\mathrm{rank}(A)$ if it is known:
(I) $A^4 = 0$
(II) $A^3 = 0$
(III) $A^2 = 0$

Now, I am very new to Jordan Forms and this is related, but I have no clue whatsoever on the relationship between Jordan Form, and knowing the Rank of a matrix. All I know is that to calculate how much Jordan blocks of size $k x k$ there are involves an equation using ranks.
Any help will be appreciated!

Let $J_A$ be a JNF for $A$. You know that for some invertible $P$ you have $A=PJ_AP^{-1}$, therefore $0=A^4=PJ_A^4P^{-1}$, therefore $J_A^4=0$. Now think about what $J_A$ looks like and consider its powers, this and this. Also recall that $\operatorname{rank}(A)=\operatorname{rank}(J_A)$, (why?).
