1. ## 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.

2. 01 00
00 10
are nilpotent, but their sum,
01
10
is inversible so it is not nilpotent

3. 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.
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.