# linear algebra: one more case, when \$ A + B \$ is a nilpotent matrix, when both \$ A \$ and \$ B \$ are nilpotent

How to prove that $$A + B$$ is nilpotent, when $$A$$, $$B$$, $$[A, B]$$ they are nilpotent matrices, and also $$A$$ Y $$[A, B]$$, $$B$$ Y $$[A, B]$$ They are pairs of interchangeable matrices.

It seems that you should use a binomial formula for interchangeable matrices, but if
$$A * [A, B] = [A, B] * A \ B * [A, B] = [A, B] * B$$ it means that $$A$$ Y $$B$$ Is it interchangeable?