Rule of Sarrus

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.

Consider a 3x3 matrix $$M=egin{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=egin{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=egin{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.

References

*Gerd Fischer: Analytische Geometrie. 4-te Auflage, Vieweg 1985, ISBN 3-528-37235-4, P.145 (German)
* [http://planetmath.org/encyclopedia/RuleOfSarrus.html Sarrus' rule at Planetmath]