| Version 2 |
Version 1 |
| For calculating the value of a determinant |
For calculating the value of a determinant |
| $$D = |
$$D = |
| \left|\begin{matrix} |
\left|\begin{matrix} |
| a_{11} & a_{12} & a_{13}\\ |
a_{11} & a_{12} & a_{13}\\ |
| a_{21} & a_{22} & a_{23}\\ |
a_{21} & a_{22} & a_{23}\\ |
| a_{31} & a_{32} & a_{33} |
a_{31} & a_{32} & a_{33} |
| \end{matrix}\right| |
\end{matrix}\right| |
| $$ |
$$ |
| with three rows, it is comfortable to use the {\em rule of Sarrus}. |
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). |
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} |
\left|\begin{matrix} |
| a_{11} & a_{12} & a_{13}\\ |
a_{11} & a_{12} & a_{13}\\ |
| a_{21} & a_{22} & a_{23}\\ |
a_{21} & a_{22} & a_{23}\\ |
| a_{31} & a_{32} & a_{33} |
a_{31} & a_{32} & a_{33} |
| \end{matrix}\right| |
\end{matrix}\right| |
| \begin{matrix} |
\begin{matrix} |
| \,a_{11} & a_{12}\\ |
\,a_{11} & a_{12}\\ |
| \,a_{21} & a_{22}\\ |
\,a_{21} & a_{22}\\ |
| \,a_{31} & a_{32} |
\,a_{31} & a_{32} |
| \end{matrix} |
\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 the second diagonal of $D$. Accordingly, one obtains the expression
|
Then one sums the products on all lines parallel to the main diagonal of $D$ and subtracts the products on the lines parallel 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_{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},$$ |
-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33},$$ |
| which gives the value of the determinant $D$. |
which gives the value of the determinant $D$. |
|
|
| There is no corresponding rule for determinants with more or less rows. |
There is no corresponding rule for determinants with more or less rows. |
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