If we have a matrices
$$A=\begin{bmatrix} a & b\\ c & d\end{bmatrix}
\hskip5mm \mbox{ and } \hskip5mm
e_{12}(\lambda)=
\begin{bmatrix}
1 & \lambda \\
0 & 1
\end{bmatrix}
$$
then by doing product
$$
Ae_{12}(\lambda) =
\begin{bmatrix}
a & a\lambda + b\\
c & c\lambda + d
\end{bmatrix} \hskip5mm \mbox{ and } \hskip5mm
e_{12}(\lambda)A =
\begin{bmatrix}
a +c\lambda & b+d\lambda \\
c & d
\end{bmatrix}
$$
we can interpret that right multiplication by $e_{12}$ to $A$ gives a column-operation: add $\lambda$-times first column to the second column.
In similar way, left multiplication by $e_{12}(\lambda)$ to $A$ gives row-operation on $A$.
Question: Is there any conceptual (not computational, if any) way to see that elementary row and column operations on a matrix can be expressed as multiplication by elementary matrices on left or right, accordingly?
Best Answer
Let $A,B$ be two matrices of order $n$.
We can describe $B$ as $B=(b_1,\ldots,b_n)$, where $b_i$ is its column $i$.
Notice that $AB=(Ab_1,\ldots,Ab_n)$.
Now apply some column-operation on $AB$.
For example, let's say that its column $i$ is multiplied by $\lambda$.
So we obtain $(Ab_1,\ldots,\lambda Ab_i,\ldots,Ab_n)$.
Notice that $(Ab_1,\ldots,\lambda Ab_i,\ldots,Ab_n)=A(b_1,\ldots,\lambda b_i,\ldots,b_n)$.
So in order to apply some column-operation on $AB$, we can first apply it on $B$ and then multiply the resulting matrix with $A$.
Can you see what happens if $B=Id$?
The same reasoning can be used for row-opperations.