rule of SarrusRuleOfSarrus2007-09-20 17:03:322008-03-16 12:56:58Resultdeterminantmnemonic% this is the default PlanetMath preamble. as your knowledge
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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 {\em 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 \PMlinkescapetext{right side} 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.