Rule of Sarrus

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Sarrus' rule: solid diagonals - dashed diagonals

Sarrus' rule or Sarrus' scheme is a method and a memorization scheme to compute the determinant of a 3×3 matrix. It is named after the French mathematician Pierre Frédéric Sarrus.

Consider a 3×3 matrix

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

then its determinant can be computed by the following scheme:

Write out the first 2 columns of the matrix to the right of the 3rd column, so that you have 5 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:

\det(M) =\begin{vmatrix} a_{11} & a_{12} & a_{13} \\  a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} = 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}.

A similar scheme based on diagonals works for 2x2 matrices:

\det(M)=\begin{vmatrix} a_{11} & a_{12}  \\  a_{21} & a_{22} \end{vmatrix} = 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 by looking at the Laplace expansion of a 3×3 matrix.

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