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Rule of Sarrus

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(Redirected from Sarrus's rule) Mnemonic device for calculating 3 by 3 matrix determinants
Rule of Sarrus: The determinant of the three columns on the left is the sum of the products along the down-right diagonals minus the sum of the products along the up-right diagonals.

In matrix theory, the rule of Sarrus is a mnemonic device for computing the determinant of a 3 × 3 {\displaystyle 3\times 3} matrix named after the French mathematician Pierre Frédéric Sarrus.

Consider a 3 × 3 {\displaystyle 3\times 3} matrix

M = [ a b c d e f g h i ] {\displaystyle M={\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}}}

then its determinant can be computed by the following scheme.

Write out the first two columns of the matrix to the right of the third column, giving five columns in a row. Then add the products of the diagonals going from top to bottom (solid) and subtract the products of the diagonals going from bottom to top (dashed). This yields

det ( M ) = | a b c d e f g h i | = a e i + b f g + c d h − c e g − b d i − a f h . {\displaystyle {\begin{aligned}\det(M)={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=aei+bfg+cdh-ceg-bdi-afh.\end{aligned}}}
Alternative vertical arrangement

A similar scheme based on diagonals works for 2 × 2 {\displaystyle 2\times 2} matrices:

| a b c d | = a d − b c {\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc}

Both are special cases of the Leibniz formula, which however does not yield similar memorization schemes for larger matrices. Sarrus' rule can also be derived using the Laplace expansion of a 3 × 3 {\displaystyle 3\times 3} matrix.

Another way of thinking of Sarrus' rule is to imagine that the matrix is wrapped around a cylinder, such that the right and left edges are joined.

References

  1. ^ Fischer, Gerd (1985). Analytische Geometrie (in German) (4th ed.). Wiesbaden: Vieweg. p. 145. ISBN 3-528-37235-4.
  2. Paul Cohn: Elements of Linear Algebra. CRC Press, 1994, ISBN 9780412552809, p. 69

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