rule of Sarrus

For calculating the value of a determinant

D=\left|\begin{matrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{matrix}\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{matrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{matrix}\right|\begin{matrix}\,a_{11}&a_{12}\\ \,a_{21}&a_{22}\\ \,a_{31}&a_{32}\end{matrix}

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