Matrix multiplication $AB=0$ so $A=0 $ or $B=0$

Question

I have tried with the idea of $AA^T=0 $ and use the trace, but nothing changed.

I have also to prove : if $AB=0 $ then $BA=0$ .

Answer 1

The ring of square matrices is not integral i.e if $AB=0$ then it is not necessary that one of them is the zero matrix.

take for example: $$ A= \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & z \\ \end{matrix} $$

and

$$ B= \begin{matrix} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{matrix} $$

This example also shows (for $z\neq0$) that if $AB=0$ then it does not mean necessarilly that $BA=0$ .