Nilpotent commutative matrices $A, B$ $\Longrightarrow$ $A+B$ nilpotent.

Question

How to prove that if $A, B$ are matrix of $n\times n$ nilpotents so that $AB=BA$ then $A+B$ is nilpotent.

Answer 1

If $A^m = 0$ and $B^n = 0$ then what is $(A+B)^{m+n}$? Use the binomial theorem (uses commutativity!)

Edit: I'll give you one more step. $$ (A+B)^{m+n} = \sum_{i=0}^{m+n} {m+n \choose i}A^{i}B^{m+n-i} $$ can you show that either $A^i = 0$ or $B^{m+n-i}=0$?