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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 |