Mnemonic device for calculating 3 by 3 matrix determinants
Rule of Sarrus: The determinant of the three columns on the left is the sum of the products along the down-right diagonals minus the sum of the products along the up-right diagonals.
In matrix theory, the rule of Sarrus is a mnemonic device for computing the determinant of a matrix named after the French mathematician Pierre Frédéric Sarrus.[1]
Consider a matrix
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]
Alternative vertical arrangement
A similar scheme based on diagonals works for matrices:[1]
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 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.
^ abcdFischer, Gerd (1985). Analytische Geometrie (in German) (4th ed.). Wiesbaden: Vieweg. p. 145. ISBN 3-528-37235-4.
^Paul Cohn: Elements of Linear Algebra. CRC Press, 1994, ISBN 9780412552809, p. 69
In matrix theory, the ruleofSarrus is a mnemonic device for computing the determinant of a 3 × 3 {\displaystyle 3\times 3} matrix named after the French...
6 terms (ruleofSarrus). The more lengthy Leibniz formula generalizes these two formulae to all dimensions. The determinant of a product of square matrices...
number of transpositions, so the term appears with negative sign. The ruleofSarrus is a mnemonic for the expanded form of this determinant: the sum of the...
&I&={}&(ae-bd).\\\end{alignedat}}} The determinant of A can be computed by applying the ruleofSarrus as follows: det ( A ) = a A + b B + c C . {\displaystyle...
of a and b. As explained below, the cross product can be expressed in the form of a determinant of a special 3 × 3 matrix. According to Sarrus'srule...
Library of Ashurbanipal. The Akkadian name is normalized as either Šarru-ukīn or Šarru-kēn. The name's cuneiform spelling is variously LUGAL-ú-kin, šar-ru-gen6...
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Michalowski, Piotr (2008), Brisch, N. (ed.), "The Mortal Kings of Ur: A Short Century of Divine Rule in Ancient Mesopotamia" (PDF), Religion and Power: Divine...
who ruled as king 627–612, Ninurta-sharru-usur (Ninurta-šarru-uṣur), who played no political role Libbali-sharrat was presumably the mother of Ashurbanipal's...
was able to rule in Tuwana for a very long period until at least c. 709 BC, at which date he was mentioned in the letter of Aššur-šarru-uṣur. Warpalawas...
of the Achaemenid Empire. Although Sargon of Akkad is often referred to as the "founder" of Akkad, the city itself probably existed before his rule;...
of Sidon and his vassal and oversaw the submission of many of the surrounding cities to his rule. Faced with a massive Assyrian army nearby, many of the...
interprets El as the subject, "El rules/struggles", from sarar (שָׂרַר) 'to rule' (cognate with sar (שַׂר) 'ruler', Akkadian šarru 'ruler, king'), which is likely...
marks, boxes, or other symbols instead of cuneiform script. Sargon II (Neo-Assyrian Akkadian: 𒈗𒁺, romanized: Šarru-kīn, meaning "the faithful king" or...
son of king Darius the Achaemenian. — Xerxes I of Persia The standard royal title of the Arsacid (Parthian) kings while in Babylon was Aršaka šarru ("Arsacid...
(1837) have been among the contributors. An important general work is that ofSarrus (1842) which was condensed and improved by Cauchy (1844). Other valuable...
ruler to introduce the title šarru dannu ("mighty king"), replacing the earlier epithet dannum. When the Third Dynasty of Ur collapsed and its vassals...
the majority of its existence as an independent kingdom, Babylon ruled most of southern Mesopotamia, composed of the ancient regions of Sumer and Akkad...
the ruleof Shamshi-Adad I and his successors, of Amorite descent and originally from the south, a more absolute form of kingship, inspired by that of Babylon...
territory of the kingdom of Ḫubuškia), were able to expand their territories at the expense of Assyria and capture the fortresses ofŠarru-iqbi and Dūr-Ellil...
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