In matrix theory, the rule of Sarrus is a mnemonic device for computing the determinant of a
3 x 3
Consider a
3 x 3
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]
\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 x 2
\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 x 3
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.