Rule of Sarrus

In matrix theory, the rule of Sarrus is a mnemonic device for computing the determinant of a $3\times 3$ matrix named after the French mathematician Pierre Frédéric Sarrus.^[1]

Consider a $3\times 3$ matrix

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^[1]^[2]

{\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}}

A similar scheme based on diagonals works for $2\times 2$ matrices:^[1]

{\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 $3\times 3$ matrix.^[1]

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

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

External links

Sarrus' rule at Planetmath
Linear Algebra: Rule of Sarrus of Determinants at khanacademy.org

[Fischer-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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