How to prove that $B$ is nilpotent. 
Let $A$ and $B$ be complex matrices with $AB^2-B^2A=B$. Prove that $B$ is nilpotent.

By the way: This problem is from American Mathematical Monthly, Problem 10339,and this solution post 1996 American Mathematical Monthly, page 907.
My question: This problem has other nice solutions? Thank you.


Multiply both sides by $B^{n-1}$ on the right so the given condition is that $B^n = AB^{n+1} - B^2 A B^{n-1}.$ The trace of a product is invariant under cyclic permutations of the factors, so the trace of the right hand side is zero. The trace of $B^n$ is zero for all $n\geq 1$ so $B$ is nilpotent.
