Rule of Sarrus

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 $3\times 3$ matrix named after the French mathematician Pierre Frédéric Sarrus.

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

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

Alternative vertical arrangement

A similar scheme based on diagonals works for $2\times 2$ matrices:

${\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc$

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.

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.

External links

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