matrix theory In mathematics, a matrix (: matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. ...

, the rule of Sarrus is a

mnemonic device A mnemonic device ( ), memory trick or memory device is any learning technique that aids information retention or retrieval in the human memory, often by associating the information with something that is easier to remember. It makes use of e ...

for computing the

determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...

of a

3 \times 3

matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

named after the French mathematician

Pierre Frédéric Sarrus Pierre Frédéric Sarrus (; 10 March 1798, Saint-Affrique – 20 November 1861) was a French mathematician. Sarrus was a professor at the University of Strasbourg, France (1826–1856) and a member of the French Academy of Sciences in Paris (184 ...

. Consider a

3 \times 3

matrix :

M=\begina&b&c\\d&e&f\\g&h&i\end

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 yieldsPaul Cohn: ''Elements of Linear Algebra''. CRC Press, 1994,
p. 69
/ref> :

\begin
\det(M)= \begin
a&b&c\\d&e&f\\g&h&i
\end=
aei + bfg + cdh - gec - hfa - idb .
\end

A similar scheme based on diagonals works for

2 \times 2

matrices: :

\begin
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 In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an -matrix as a weighted sum of minors, which are the determinants of some - submatrices of . Spe ...

of a

3 \times 3

matrix. 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 Linear algebra Determinants Mnemonics