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Mathematics-Online course: Linear Algebra - Matrices - Determinants

Sarrus Scheme

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As illustrated in the figure the determinant of a $ 3\times 3$-matrix is the sum of products corresponding to the different diagonals.

\includegraphics[origin=cc,width=.8\linewidth]{b_sarrus_bild_vert}

An analogon to this scheme, named after Sarrus, does not exist for higher dimensions $ n$. For $ n=4$ the determinant is already a sum of $ 24$ products.

The Sarrus scheme can easily be verified by expanding the determinant into a sum over permutations. The following table shows all addends as well as the interchanges associated with the permutations $ i$ from which the sign $ \sigma$ results.

$ i$ Interchanges $ \sigma(i)$ Addend
$ (1,2,3)$   $ +$ $ +a_{1,1}a_{2,2}a_{3,3}$
$ (2,3,1)$ $ \to(3,2,1)\to(1,2,3)$ $ +$ $ +a_{2,1}a_{3,2}a_{1,3}$
$ (3,1,2)$ $ \to(2,1,3)\to(1,2,3)$ $ +$ $ +a_{3,1}a_{1,2}a_{2,3}$
$ (3,2,1)$ $ \to(1,2,3)$ $ -$ $ -a_{3,1}a_{2,2}a_{1,3}$
$ (1,3,2)$ $ \to(1,2,3)$ $ -$ $ -a_{1,1}a_{3,2}a_{2,3}$
$ (2,1,3)$ $ \to(1,2,3)$ $ -$ $ -a_{2,1}a_{1,2}a_{3,3}$

(Authors: Burkhardt/Höllig/Hörner)
  automatically generated 4/21/2005