assume $A\in M_n(\mathbb C)$,$A^2=0$ how prove $\exists C,B\in M_n(\mathbb C)$ such that $A=BC$ and $CB=0$.
Thanks in advance
Hint: Let $C$ be projection onto a complement of the kernel of $A$ and let $B = A$. Then you should be able to show that $BC = A$ but $CB = 0$.
WLOG, you may assume that $A$ is in Jordan form. Then $A^2=0$ means that every Jordan block of $A$ is either the zero scalar or a $2\times 2$ nilpotent Jordan block. Now, Jim's answer to another question that was posted 3 hours ago is useful too.