Rule of Sarrus

In linear algebra, the Rule of Sarrus is a mnemonic device for computing the determinant of a ${\displaystyle 3\times 3}$ matrix named after the French mathematician Pierre Frédéric Sarrus.[1]

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.

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

${\displaystyle M={\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}},}$

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]

${\displaystyle {\begin{aligned}\det(M)&=\det {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}}\\[6pt]&=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{31}a_{22}a_{13}-a_{32}a_{23}a_{11}-a_{33}a_{21}a_{12}.\end{aligned}}}$

Alternative vertical arrangement

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

${\displaystyle \det(M)=\det {\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}=a_{11}a_{22}-a_{21}a_{12}.}$

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

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

External links

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