Find nilpotent

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.

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

So inversible means no nilpotent? How so? Thanks.

Quote:

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Originally Posted by tttcomrader

So inversible means no nilpotent? How so? Thanks.

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

Quote:

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Originally Posted by ThePerfectHacker

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

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