Rule of Sarrus

From Wikipedia, the free encyclopedia

Sarrus' rule or Sarrus' scheme is a method and a memorization scheme to compute the determinant of a 3x3 matrix. It is named after the French mathematician Pierre Frédéric Sarrus.

Sarrus rule: solid diagonals - dashed diagonals
Sarrus rule: solid diagonals - dashed diagonals

Consider a 3x3 matrix M=\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} , then its determinant can be computed by the following scheme:

Repeat the first 2 columns of the matrix behind the 3rd column, so that you have 5 columns in a row. Then add the products of the diagonals going from top to bottom (solid) and subtract the products of the diagonals going from bottom to top (dashed). This yields:

M=\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} = a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{31}a_{22}a_{13}-a_{32}a_{23}a_{11}-a_{33}a_{21}a_{12}

A similar scheme based on diagonals works for 2x2 matrices: M=\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = a_{11}a_{22} - a_{21}a_{12}

Both are special cases of the Leibniz formula, which however does not yield similar memorization schemes for larger matrices.

[edit] References

Languages