sarrus

Sarrus most often refers to Sarrus' rule, a mnemonic method for computing the determinant of a 3x3 matrix. The rule is named after the French mathematician Pierre Sarrus, who popularized it in the 19th century as a quick algebraic shortcut.

Consider a 3x3 matrix written as

[a b c; d e f; g h i].

To apply the rule, one can conceptually extend the matrix by repeating the first two columns to the right, forming a six-column block. The determinant is obtained by adding the products of the diagonals from the top left to the bottom right (aei, bfg, cdh) and subtracting the products of the diagonals from the bottom left to the top right (gec, hfa, idb). Equivalently, det = aei + bfg + cdh − ceg − afh − bdi.

The method provides a straightforward calculation specifically for 3x3 matrices and is commonly taught as a mnemonic in introductory linear algebra. It is not a general technique for larger matrices, where determinant computation relies on expansion by cofactors, row reduction, or LU decomposition. While Sarrus' rule can be convenient for hand calculations, it is primarily of historical and educational value in illustrating diagonal products and the determinant concept. Modern computational practice typically uses more systematic algorithms that scale to larger systems and improve numerical stability.