Translate [[elementary operations on a linear system|elementary operations on the linear system]] into operations on the rows of the [[augmented matrix]]:

  - 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 $i\ne j$.

  * 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.