Elementary row operations versus matrices

Can every elementary row op on a matrix be represented by multiplying that very matrix by an elementary matrix?

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Yes. Every elementary row operation can be performed by left multiplication by an elementary matrix. Just as every elementary column operation can be performed by right multiplication by an elementary matrix. – Bill Cook Nov 5 '11 at 23:04

The answer to your question is yes. I will give you the matrices in each case: Let $M$ be an $n \times n$ matrix.

1) Multiplying row $i$ of $M$ by a non-zero scalar $c$ corresponds to multiplying $M$ on the left by the matrix $X$ where $X$ is the same as the identity matrix, but the $(i,i)^{th}$ entry replaced by $c$.

2) Interchanging rows $i$ and $j$ of $M$ corresponds to multiplying $M$ on the left by the matrix $X_1$ where $X_1$ is the identity matrix with the $i^{th}$ and $j^{th}$ diagonal entries replaced by $0$ and the $(i,j)^{th}$ and $(j,i)^{th}$ entries replaced by $1$.

3) Adding $c(row \hspace{1mm}j)$ to $row \hspace{1mm} i$ corresponds to multiplying $M$ on the left by the matrix $X_2$ where $X_2$ is the identity matrix with $(i,j)^{th}$ entry replaced by $c$.

Hope this helps. Column operations correspond to multiplying by elementary matrices on the right.

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I hope there are no errors with how I wrote the entries. – Rankeya Nov 5 '11 at 23:09
+1: This is very helpful! :) – kid Nov 5 '11 at 23:12
Note also that the elementary matrix corresponding to a given row operation can be obtained by doing that row operation to the identity matrix of the correct size, and likewise for column operations. – Brad Nov 5 '11 at 23:32