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If you have ever drawn six slanting lines across a determinant in a Class 12 exam, you have used an idea from a 19th-century French professor. Pierre Frédéric Sarrus gave students a picture for the six products inside every 3×3 determinant. Here is who he was, how the rule works, and the one trap it sets.
Who was Pierre Frédéric Sarrus?
Sarrus was born on 10 March 1798 in Saint-Affrique, in the Aveyron region of southern France, and died there on 20 November 1861, according to the French biography on BibM@th. The same biography says he was refused the certificate he needed to enter medical school because he openly showed his Bonapartist views and his Protestant faith. He went on to earn a doctorate in mathematics in 1821 and the agrégation in 1822.
After teaching in Pézenas and Perpignan, Sarrus became a mathematics professor at the University of Strasbourg in 1829 and served as dean of its science faculty from 1839 to 1852 (BibM@th). His main research was the calculus of variations, the mechanics of jointed parts and the theory of equations. A memoir of his on the calculus of variations won the Grand Prize of the French Academy of Sciences in 1843.
He also built something you can still see today. This University of Edinburgh page on exact straight-line linkages says the first such linkage was described by Sarrus in 1853, and that its parts move in three dimensions. It is used widely in jacks and lifting platforms.
What is the rule of Sarrus for a 3×3 determinant?
The rule of Sarrus is a memory device for the determinant of a 3×3 matrix: copy the first two columns to the right, add the products along the three downward diagonals, and subtract the products along the three upward diagonals (Wikipedia: Rule of Sarrus). The French maths magazine Tangente describes the same procedure and calls it a memory aid reserved for 3×3 matrices.
For a matrix with rows (a, b, c), (d, e, f) and (g, h, i), the rule gives:
det = aei + bfg + cdh − gec − hfa − idb
That is exactly what expanding along the first row gives; the rule just lays the six terms out where your eye can follow them. To see why the signs come out this way, read our post on finding minors and cofactors of a 3×3 matrix, which walks through the sign checkerboard.
How do you use the rule of Sarrus? A worked example
Take the matrix with rows (2, −1, 3), (0, 4, 1) and (5, 2, −2).
Copy the first two columns to the right. The rows become 2, −1, 3, 2, −1 and 0, 4, 1, 0, 4 and 5, 2, −2, 5, 2.
Downward diagonals: 2 × 4 × (−2) = −16; (−1) × 1 × 5 = −5; 3 × 0 × 2 = 0. Their sum is −21.
Upward diagonals: 5 × 4 × 3 = 60; 2 × 1 × 2 = 4; (−2) × 0 × (−1) = 0. Their sum is 64.
Determinant = (sum down) − (sum up) = −21 − 64 = −85.
Check it by cofactor expansion along row 1: 2(−8 − 2) − (−1)(0 − 5) + 3(0 − 20) = −20 − 5 − 60 = −85. Same answer.
If you keep slipping on signs, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer, so you can see the exact line where your sign flipped.
Does the rule of Sarrus work for a 4×4 matrix?
No. Tangente states that for a matrix larger than 3×3 the diagonal scheme does not extend, and Wikipedia notes that the general determinant formula gives no similar memory scheme for larger matrices. For 4×4 and larger, reduce with row operations or expand by cofactors.
The reason is counting. A 3×3 determinant has 3! = 6 terms, and the six diagonals catch all of them. A 4×4 determinant has 4! = 24 terms, but copying columns and drawing diagonals gives you only 8 products. The other 16 are simply missing, so a diagonal answer for a 4×4 is wrong even if every multiplication is right.
When should you use Sarrus in an exam?
Check this before you start:
Plain 3×3 with numbers: Sarrus is fast and hard to get wrong if you keep the two sums separate.
3×3 with a row or column full of zeros: expand along that row; it is shorter.
3×3 with letters or a pattern: simplify with row operations first, then expand.
4×4 or larger: never Sarrus; reduce with row operations or expand by cofactors.
Determinants turn up well beyond matrices chapters. You will use the same 3×3 expansion in Cramer's rule for three variables and in the cross product written as an i, j, k determinant.
If the idea of a determinant itself still feels shaky, the Fermi Concept Library gives each mathematics concept its own lesson, so you can fill the gap before you practise the shortcut.
