Translate elementary operations on the linear system into operations on the rows of the augmented matrix:

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 $i\ne j$.

• These operations are called elementary row operations or EROs on the matrix.
• The systems of linear equations corresponding to these matrices have exactly the same solutions.