TITLE: Are these two equalities equivalent: $ABA=0$ and $BA=0$?
QUESTION [5 upvotes]: There are two equalities as follows:

$ABA=0$ , $BA=0$
where $A$ and $B$ are two $n\times n$ matrices.

Are the mentioned equalities the same? if not, why?
Thanks in advance.

REPLY [4 votes]: If these two equations would be the same then also for $B=A$ the equations $A^3=0$ and $A^2=0$ should be equivalent what certainly is not the case for all matrices,    
consider for example  the case of matrix   
$A= \begin{bmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0  \end{bmatrix}$,
$A^2= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0  \end{bmatrix}$,
$A^3= \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0  \end{bmatrix}$.