matrices - left multiplication by invertible matrix doesn't change reduced row echelon form

How to proof that for the matrices:

$A=\begin{pmatrix}
1 & 0 \\
a & 1

\end{pmatrix}
$ and $B=\begin{pmatrix}
2 & 0 \\
0 & 3
\end{pmatrix}
$,
$X\in M_{23}(\mathbb R)$

$A(BX)$ has the same reduced row echelon form as $X$ ?

Of course I know: $A(BX) <=> (AB)X$, A is an elementary matrix and B is the product of an elementary matrix. I also know, that the reduced row echelon form is distinct. But how to show that ?