Find nilpotent

**tttcomrader** · November 8th 2008, 11:57 AM

Find two nilpotent 2 by 2 matrices such that their sum is not nilpotent.

I have been trying a few matrices but none are working, is there a more logical, easier way to find these matrices? thanks.

**clic-clac** · November 8th 2008, 12:03 PM

01 00
00 10
are nilpotent, but their sum,
01
10
is inversible so it is not nilpotent

**tttcomrader** · November 8th 2008, 04:50 PM

So inversible means no nilpotent? How so? Thanks.

**ThePerfectHacker** · November 8th 2008, 05:10 PM

Originally Posted by tttcomrader

So inversible means no nilpotent? How so? Thanks.

Because if $A$ is invertible then $A^n$ is invertible. Thus, if $A^n$ is not invertible then $A$ is not invertible. If $A^n$ is zero matrix then it means $A^n$ is not invertible and so $A$ is not invertible.

**arbolis** · January 27th 2009, 07:22 AM

Originally Posted by ThePerfectHacker

Because if $A$ is invertible then $A^n$ is invertible. Thus, if $A^n$ is not invertible then $A$ is not invertible. If $A^n$ is zero matrix then it means $A^n$ is not invertible and so $A$ is not invertible.

Thank you TPH, you just answered a question of mine. I'm glad I've searched into the forum.

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