Rule of Sarrus

In the linear algebra which is usually Sarrus (also sarrussche rule or picket fence rule ) a method with which the determinant a - matrix can be easily calculated. This rule is named after the French mathematician Pierre Frédéric Sarrus . It is a special case of the Leibniz formula . ${\ displaystyle 3 \ times 3}$

application

For the matrix ${\ displaystyle 3 \ times 3}$

{\ displaystyle A = {\ begin {pmatrix} a _ {\ text {11}} & a _ {\ text {12}} & a _ {\ text {13}} \\ a _ {\ text {21}} & a _ {\ text { 22}} & a _ {\ text {23}} \\ a _ {\ text {31}} & a _ {\ text {32}} & a _ {\ text {33}} \ end {pmatrix}}}

the determinant consists of 6 summands of 3 factors each, which can easily be determined with the following scheme.

You write the first two columns of the matrix to the right of the matrix and form products of 3 numbers each, which are connected by the oblique lines. Then the products running from top left to bottom right are added and the products running from bottom left to top right are subtracted. Another common practice is to append the first two rows to the bottom of the matrix and then follow the pattern in the illustration above. In this way we get the determinant of : ${\ displaystyle A}$

{\ displaystyle \ det (A) = a _ {\ text {11}} a _ {\ text {22}} a _ {\ text {33}} + a _ {\ text {12}} a _ {\ text {23}} a _ {\ text {31}} + a _ {\ text {13}} a _ {\ text {21}} a _ {\ text {32}} - a _ {\ text {13}} a _ {\ text {22}} a _ {\ text {31}} - a _ {\ text {11}} a _ {\ text {23}} a _ {\ text {32}} - a _ {\ text {12}} a _ {\ text {21}} a _ {\ text {33}}}

The rule, which looks similar, applies to matrices ${\ displaystyle 2 \ times {2}}$

{\ displaystyle \ det (A) = {\ begin {vmatrix} a_ {11} & a_ {12} \\ a_ {21} & a_ {22} \ end {vmatrix}} = a _ {\ text {11}} a_ { \ text {22}} - a _ {\ text {12}} a _ {\ text {21}}}

Sarrus' rule only applies to third-order determinants. For more than three dimensions, the Leibniz formula quickly becomes very large, the computational effort increases with the faculty of the dimension. In the presence of many zero entries can Laplace expansion theorem simplify the calculation. Substantially faster calculation options, even in the general case, are offered by decomposing the matrix, for example using the Gaussian algorithm .

literature

Gerd Fischer: Analytical Geometry. 4th edition. Vieweg, 1985, ISBN 3-528-37235-4 , p. 145.

Web links

Commons : Sarrus rule - collection of pictures, videos and audio files

rule of Sarrus. On: planetmath.org.
Sarrus rule. On: mathematik.net.