Rule of Sarrus

In matrix theory, 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

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

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]

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

Alternative vertical arrangement

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

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

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 ${\displaystyle 3\times 3}$ 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.