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Elementary row operations:

1) Interchange any two rows of the matrix

2) Multiply every entry of some row of the matrix by the same nonzero scalar

3) Add a multiple of one row of the matrix to another row

Elementary Matrix:

An $n\times n$ matrix that is produced by performing exactly one elementary row operation on an $n\times n$ identity matrix.

The other matrix in the multiplication has a dimension of $n\times m$.

e.g. Multiply the first matrix to the second matrix:

\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{pmatrix} Output: \begin{pmatrix} 1 & 2 & 3 \\ 5 & 7 & 9 \\ 7 & 8 & 9 \\ \end{pmatrix}

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    $\begingroup$ I suggest you take a look at Artin's Algebra text book. The first chapter has a discussion on elementary matrices as far as I can recall. $\endgroup$ – r9m Jul 7 '15 at 6:47
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Here's an attempt at a conceptual proof.

If it's true that a row operation can be performed by multiplying by some matrix $ M$, then $ M = M I $ is the matrix you get by performing the row operation on the identity matrix.

The only question is, can an elementary row operation be performed by multiplying by some matrix $ M $.

The answeris yes: A row operation takes a matrix $ A $ as input and applies the same linear transformation $ T $ to each column of $ A $. This is the same as multiplying each column of $ A $ by $ M $, where $ M$ is the matrix that represents $ T$. And that's the same as multiplying $ A $ by $ M $.

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