Sarrus' Rule¶

Method¶

Sarrus’ rule is a mnemonic for computing the determinant of a $3\times3$ matrix $\boldsymbol{A}=(a_{ij})$ $(a_{ij})$ : write the first two columns of $\boldsymbol{A}$ again to the right of the matrix, add the products along the three falling diagonals and subtract the products along the three rising diagonals,

\det \boldsymbol{A}={}a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31} +a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31} -a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}.

Interpretation¶

Sarrus' rule is a quick way to write down the six products that make up the determinant of a $3\times3$ matrix, without forming any minors.

A 3 by 3 matrix with its first two columns copied to the right, and three diagonal lines whose products are added and three anti-diagonal lines whose products are subtracted.

With the first two columns copied to the right, the three solid lines run downwards to the right and give the products that are added, $a_{11}a_{22}a_{33}$ , $a_{12}a_{23}a_{31}$ and $a_{13}a_{21}a_{32}$ ; the three dashed lines run upwards to the right and give the products that are subtracted. Each product uses one entry from every row and every column.

The rule is cofactor expansion along the first row written out,

\det \boldsymbol{A}={}a_{11}(a_{22}a_{33}-a_{23}a_{32}) -a_{12}(a_{21}a_{33}-a_{23}a_{31}) +a_{13}(a_{21}a_{32}-a_{22}a_{31}),

which multiplies out to the same six terms with the same signs.

The rule works only for $3\times3$ matrices: copying columns of a $4\times4$ matrix in the same way gives just $8$ diagonal products, while its determinant consists of $24$ such products.

Key Results¶

The three falling diagonal products receive positive signs.
The three rising diagonal products receive negative signs.

Examples¶

For $\boldsymbol{I}_3$ , Sarrus’ rule gives $1$ .
For a matrix with two equal rows, Sarrus’ rule gives determinant $0$ .

Prerequisites¶

Determinant

Sources¶

Math II: Linear Algebra · Week 4

Aliases: rule of Sarrus, Sarrus scheme, Sarrus's rule

Last updated: 22 October 2026 19:04