Let

$\displaystyle e$ be an elementary row operation,

$\displaystyle A,B,C$ be a $\displaystyle n \times n$ matrices

By $\displaystyle e(A)$ we mean the resultant matrix after we perform $\displaystyle e$ on $\displaystyle A$

Does the the following hold true -

$\displaystyle e(A)B = Ae(B)$ ?

Also, we know the $\displaystyle e(A)=e(I_n)A$. One way to prove this is for each kind of elementary row operation establish this fact by performing the computation on the matrix elements.

I was wondering if there is a more conceptual/abstract argument, logic, reason behind this to work.

Thanks