$A+B$ and $AB$ are nilpotent matrices, are $A$ and $B$ nilpotent?

Question

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!

Answer 1

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