mathematics

Rule of Sarrus

Rule of Sarrus is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Rule of Sarrus rather than just read about it. In short: In matrix theory, the rule of Sarrus is a mnemonic device for computing the determinant of a 3 × 3 {\displaystyle 3\times 3} matrix named after the French mathematician Pierre Frédéric Sarrus. Consider a 3 × 3 {\displaystyle 3\times 3} matrix M = [ a b c d e f g h i ] {\displaystyle M={\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}}} then its determinant can be computed by the following scheme.

Key takeaways

Rule of Sarrus belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Rule of Sarrus to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Rule of Sarrus from memory before moving on to harder problems.

Reference excerpt

In matrix theory, the rule of Sarrus is a mnemonic device for computing the determinant of a 3 × 3 {\displaystyle 3\times 3} matrix named after the French mathematician Pierre Frédéric Sarrus. Consider a 3 × 3 {\displaystyle 3\times 3} matrix

M = [ a b c d e f g h i ] {\displaystyle M={\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}}}

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

det ( M ) = | a b c d e f g h i | = a e i + b f g + c d h − g e c − h f a − i d b . {\displaystyle {\begin{aligned}\det(M)={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=aei+bfg+cdh-gec-hfa-idb.\end{aligned}}}

A similar scheme based on diagonals works for 2 × 2 {\displaystyle 2\times 2} matrices:

| a b c d | = a d − b c {\displaystyle {\begin{vmatrix}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. The determinants of 4x4 matrices can be found by expanding the most convenient row or column by cofactors and then applying the rule to each of the resulting determinants. (especially if there are elements in the matrix that are zero). For matrices beyond 4x4 it becomes impractical except for specific cases where there are many zero elements). Sarrus' rule can also be derived using the Laplace expansion of a 3 × 3 {\displaystyle 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

Illustrations

Worked examples

Example 1 — a first encounter with Rule of Sarrus

Start with the simplest possible case. Write down what Rule of Sarrus claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Rule of Sarrus before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Rule of Sarrus ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Rule of Sarrus

In research: Rule of Sarrus appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Rule of Sarrus in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Rule of Sarrus is common in secondary-school and first-year university syllabi. It links to neighbouring topics Determinants, Linear algebra, Mnemonics, so understanding it makes those chapters shorter.
In everyday life: Look for Rule of Sarrus outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Rule of Sarrus in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Rule of Sarrus means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Rule of Sarrus out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Rule of Sarrus in simple terms?

Why does Rule of Sarrus matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Rule of Sarrus?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Rule of Sarrus.