# Prove that multiplying an elementary matrix to a matrix can produce the same effect as an elementary row operation.

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}

• 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. – r9m Jul 7 '15 at 6:47

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 $M$.