Rule of Sarrus
In matrix theory, the Rule of Sarrus is a mnemonic device for computing the determinant of a [math]\displaystyle{ 3 \times 3 }[/math] matrix named after the French mathematician Pierre Frédéric Sarrus.[1]
Consider a [math]\displaystyle{ 3 \times 3 }[/math] matrix
- [math]\displaystyle{ M=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix} }[/math]
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]
- [math]\displaystyle{ \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} }[/math]
A similar scheme based on diagonals works for [math]\displaystyle{ 2 \times 2 }[/math] matrices:[1]
- [math]\displaystyle{ \begin{vmatrix} a&b\\c&d \end{vmatrix} =ad - bc }[/math]
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 [math]\displaystyle{ 3 \times 3 }[/math] matrix.[1]
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.
References
External links
- Sarrus' rule at Planetmath
- Linear Algebra: Rule of Sarrus of Determinants at khanacademy.org
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