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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!

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TheNotMe
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    The Jordan form shows that if $[A]{ii} = \lambda$, then $[A^k]{ii} = \lambda^k$. So all the above conditions imply that the diagonal of $A$ is zero. Then the conditions give restrictions on the size of the Jordan blocks. Figure out what this means for the rank. – copper.hat Jun 04 '13 at 19:08

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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?).

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Git Gud
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  • All I could realize is that all the $\lambda$ are $0$. – TheNotMe Jun 04 '13 at 20:20
  • @TheNotMe That is true, but you don't need JNF for that. Try to figure what does $J_A$ look like. There are multiple looks it might get. You'll have to check case by case, but since it's only a $5\times 5$ matrix. There aren't that many. I suggest you start by assuming WLOG that $J_A$ has its jordan blocks in descending (or ascending) order with respect to their size. – Git Gud Jun 04 '13 at 20:31
  • @TheNotMe Do you need extra help with this? – Git Gud Jun 09 '13 at 10:26
  • @TheNotMe Anytime. – Git Gud Jun 09 '13 at 15:32