# rule of Sarrus

For calculating the value of a determinant^{}

$$D=\left|\begin{array}{ccc}\hfill {a}_{11}\hfill & \hfill {a}_{12}\hfill & \hfill {a}_{13}\hfill \\ \hfill {a}_{21}\hfill & \hfill {a}_{22}\hfill & \hfill {a}_{23}\hfill \\ \hfill {a}_{31}\hfill & \hfill {a}_{32}\hfill & \hfill {a}_{33}\hfill \end{array}\right|$$ |

with three rows, it is comfortable to use the rule of Sarrus (invented 1833 by the French mathematician P. F. Sarrus).

The rule comprises that first one writes the two first columns of the determinant on the of the determinant (seeing thus a $3\times 5$ matrix!):

$$\left|\begin{array}{ccc}\hfill {a}_{11}\hfill & \hfill {a}_{12}\hfill & \hfill {a}_{13}\hfill \\ \hfill {a}_{21}\hfill & \hfill {a}_{22}\hfill & \hfill {a}_{23}\hfill \\ \hfill {a}_{31}\hfill & \hfill {a}_{32}\hfill & \hfill {a}_{33}\hfill \end{array}\right|\begin{array}{cc}\hfill {a}_{11}\hfill & \hfill {a}_{12}\hfill \\ \hfill {a}_{21}\hfill & \hfill {a}_{22}\hfill \\ \hfill {a}_{31}\hfill & \hfill {a}_{32}\hfill \end{array}$$ |

Here one sums the products on all lines parallel to the main diagonal of $D$ and subtracts the products on the lines parallel to the second diagonal of $D$. Accordingly, one obtains the expression

$${a}_{11}{a}_{22}{a}_{33}+{a}_{12}{a}_{23}{a}_{31}+{a}_{13}{a}_{21}{a}_{32}-{a}_{13}{a}_{22}{a}_{31}-{a}_{11}{a}_{23}{a}_{32}-{a}_{12}{a}_{21}{a}_{33},$$ |

which gives the value of the determinant $D$.

There is no corresponding rule for determinants with more or less rows.

Title | rule of Sarrus |
---|---|

Canonical name | RuleOfSarrus |

Date of creation | 2013-03-22 17:32:49 |

Last modified on | 2013-03-22 17:32:49 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Result |

Classification | msc 15A15 |

Synonym | Sarrus rule |

Synonym | Sarrus’ rule |

Related topic | LaplaceExpansion |