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1) we know that some degree multiplication(exponentiation) of the nilpotent matrices become zero. then what happens if we do multiplication like $ ABCDEAP$? when does this become zero matrix? ( A,B,C,D,E,P are nilpotent matrices.

2) which subsets of nilpotent matrices commute?

can anyone provide me requirement for matrices to commute or anticommute and nilpotent matrices to commut/anticommute?

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Take $A=\begin{bmatrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix} \quad\mbox{and}\quad B=\begin{bmatrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}$ –  wj32 Oct 26 '12 at 10:35
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(1) Please do not make such radical changes to your question without stating them as edits. (2) It is not true that squaring a nilpotent matrix gives you the zero matrix. (3) What are $A,B,C,D,E,P$? –  wj32 Oct 26 '12 at 10:48
    
Of course, this heavily depends on the matrices involved. –  rschwieb Oct 26 '12 at 12:43
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closed as not a real question by fpqc, Thomas, rschwieb, Norbert, J. M. Oct 27 '12 at 3:26

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1 Answer

Regarding question (1): If $A=\left(\begin{matrix}0&1\\0&0\end{matrix}\right)$ and $B=\left(\begin{matrix}0&0\\1&0\end{matrix}\right)$, then $A$ and $B$ are nilpotent, but all products of the form $(AB)^n$ are nonzero. Thus the product of nilpotent matrices may never become zero.

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