In linear algebra, the Rule of Sarrus is a

mnemonic device A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding. Mnemonics make use of elaborative encoding, retrieval cues, and imagery ...

for computing the

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...

of a

3 \times 3

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

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 (18 ...

. Consider a

3 \times 3

matrix :

M=\begin a_ & a_ & a_ \\  a_ & a_ & a_ \\ a_ & a_ & a_ \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> :

&= a_a_a_+a_a_a_+a_a_a_-a_a_a_- a_a_a_-a_a_a_. \end

A similar scheme based on diagonals works for

2 \times 2

matrices: :

\det(M)= \det \begin a_ & a_  \\  a_ & a_ \end = a_a_ - a_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 using the Laplace expansion 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