Rule of Sarrus

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

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.