# 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 linear algebra, 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

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

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

$\det(M)=\det {\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}=a_{11}a_{22}-a_{21}a_{12}.$ Both are special cases of the Leibniz formula, which however does not yield similar memorization schemes for larger matrices. Sarrus' 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.