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 a linear system|elementary operations on the linear system]] into corresponding operations on the rows of the matrix. We get three different types:

  - change the order of the rows of the matrix;
  - multiply one of the rows of the matrix by a non-zero real number; 
  - 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.