# Thread: A question on a nilpotent matrix

1. ## A question on a nilpotent matrix

Let N be an n*n matrix on a field F with nilpotency index n i.e. $\displaystyle N^n=0$ but $\displaystyle N^{n-1}\neq 0$ . Now show that N doesn't have square root i.e. there is no matrix like A where $\displaystyle A^2=N$.

2. ## Re: A question on a nilpotent matrix

Since the conclusion is negative, that is, "there does not exist", an indirect proof is indicated. Suppose the conclusion where false. That is, suppose there exist A such that $\displaystyle A^2= N$. Then $\displaystyle A^{2n}= N^n= 0$. Now, what does that say about $\displaystyle A^n$?

3. ## Re: A question on a nilpotent matrix

$\displaystyle A^n=0$? And then for $\displaystyle n \geq 2$, $\displaystyle A^{2(n-1)}=0$ , a contradiction!

4. ## Re: A question on a nilpotent matrix

Hi Human,
What you say is certainly true, but do you know why it follows that An=0 from A2n=0?

5. ## Re: A question on a nilpotent matrix

If some power of A is zero then its nth power is zero.