[Math] Let $A$ and $B$ be square matrices of the same size such that $AB = BA$ and $A$ is nilpotent. Show that $AB$ is nilpotent.

Question

A square matrix $A$ is called nilpotent is $A^k=0$ for some positive integer $k$.

Let $A$ and $B$ be square matrices of the same size such that $AB = BA$ and $A$ is nilpotent. Show that $AB$ is nilpotent.

Following that is $AB$ not equal to $BA$ in part (b), must $AB$ be nilpotent?

I don't get how to prove this. Since $A$ can represent any matrix. I have no idea how to show it.

Answer 1

Hint: if $AB=BA$ then $$(AB)^k=A^kB^k\ .$$

For the second part (with $AB\ne BA$) try $$A=\pmatrix{1&-1\cr1&-1\cr}\ ,\quad B=\pmatrix{4&3\cr2&1\cr}\ .$$