# Rule of Sarrus

Sarrus's rule: The determinant of the three columns on the left is the sum of the products along the solid diagonals minus the sum of the products along the dashed diagonals

Sarrus' rule or Sarrus' scheme is a method and a memorization scheme to compute the determinant of a 3×3 matrix. It is named after the French mathematician Pierre Frédéric Sarrus.[1]

Consider a 3×3 matrix

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

then its determinant can be computed by the following scheme:

Write out the first 2 columns of the matrix to the right of the 3rd column, so that you have 5 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)&={\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}}\\&=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 2×2 matrices:[1]

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

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 by looking at the Laplace expansion of a 3×3 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. ^ a b c d 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