Recall that when we form the augmented matrix of a linear system, each equation in the system becomes a row of the matrix. So we can translate the elementary operations on the linear system into corresponding operations on the rows of the matrix. We get three different types:

1. change the order of the rows of the matrix;
2. multiply one of the rows of the matrix by a non-zero real number;
3. replace row $j$ by “row $j$ ${}+{}$ $c\times {}$ (row $i$)”, where $c$ is a non-zero real number and $i\ne j$.

The system of linear equations corresponding to these matrices will then have exactly the same solutions.

We call these operations elementary row operations or EROs on the matrix.