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Viewing Version
3
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'rule of Sarrus'
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| Title of object: |
rule of Sarrus |
| Canonical Name: |
RuleOfSarrus |
| Type: |
Result |
| Created on: |
2007-09-20 17:03:32 |
| Modified on: |
2007-09-21 03:41:07 |
| Classification: |
msc:15A15 |
| Keywords: |
mnemonic |
Revision comment (for changes between this and next version):
| Changes for correction #13102 ('Confusion'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
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Content:
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}.
This comprises that first one writes the two first columns of the determinant on the right side of the determinant (getting thus a $3\!\times\!5$ matrix):
$$
\left|\begin{matrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{matrix}\right|
\begin{matrix}
\,a_{11} & a_{12}\\
\,a_{21} & a_{22}\\
\,a_{31} & a_{32}
\end{matrix}
$$
Then 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.
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