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Elementary row operations are performed on the augmented matrix of a system of linear equations. These operations produce a new augmented matrix corresponding to a new equivalent system of linear equations. Algorithms that use matrix elementary row operations to solve systems of linear equations are Gaussian elimination with back-substitution and Gauss-Jordan elimination.

There are three elementary row operations:

1) The interchanging of two rows,

2) The multiplying a row by a non-zero constant, and

3) The adding of a multiple of a row to another row.

Interchanging of Two Rows:

3×4 | Column 1 | Column 2 | Column 3 | Column 4 |

Row 1: | 3 | 4 | 5 | 6 |

Row 2: | -2 | 1 | 0 | 3 |

Row 3: | 4 | 2 | 2 | 4 |

Interchanged Rows 1 and 2:

3×4 | Column 1 | Column 2 | Column 3 | Column 4 |

Row 1: | -2 | 1 | 0 | 3 |

Row 2: | 3 | 4 | 5 | 6 |

Row 3: | 4 | 2 | 2 | 4 |

Multiplying a Row by a Non-zero Constant:

3×4 | Column 1 | Column 2 | Column 3 | Column 4 |

Row 1: | -4 | 0 | 2 | 6 |

Row 2: | 2 | -4 | 4 | 1 |

Row 3: | -4 | 0 | 2 | 6 |

Multiply Row 2 by 1/2:

3×4 | Column 1 | Column 2 | Column 3 | Column 4 |

Row 1: | -4 | 0 | 2 | 6 |

Row 2: | 1 | -2 | 2 | 1/2 |

Row 3: | -4 | 0 | 2 | 6 |

Adding a Multiple of a Row to Another Row:

3×4 | Column 1 | Column 2 | Column 3 | Column 4 |

Row 1: | 1 | 2 | -6 | 3 |

Row 2: | 0 | 3 | -2 | -1 |

Row 3: | 5 | 2 | 1 | -2 |

Add -2 times the First Row to Third Row:

3×4 | Column 1 | Column 2 | Column 3 | Column 4 |

Row 1: | 1 | 2 | -6 | 3 |

Row 2: | 0 | 3 | -2 | -1 |

Row 3: | 3 | -2 | 13 | -8 |