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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

closed as not a real question by BenjaLim, Thomas, rschwieb, Norbert, J. M. Oct 27 '12 at 3:26

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, see the FAQ.

1 Answer

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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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