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Suppose $A+B$ and $AB$ are nilpotent matrices, I'm thinking whether $A$ and $B$ are nilpotent.

If $A$ and $B$ transform $A$ and $B$ into Jordan form at the same time, then we can just calculate the Jordan block to see that $A$ and $B$ aren't nilpotent.

But the general case is $A$ and $B$ cannot be turned into Jordan form at the same time, what can we do? Any help would be appreciated!

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1 Answer 1

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No. Consider $A=\pmatrix{1&0\\ 0&0}$ and $B=\pmatrix{0&-1\\ 1&-1}$ for instance.

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