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Determination of the Sign of a Single Term of a Determinant

Published online by Cambridge University Press:  15 September 2014

Thomas Muir
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Extract

(1) As is well known, the first rule given for ascertaining the sign of a single term of a determinant was made known by Cramer in his Introduction à l'Analyse des Lignes Courbes algèbriques, published at Geneva in 1750. On page 658 he says—

“On donne a ces termes les signes + ou −, selon la Règle suivante. Quand un exposant est suivi dans le même terme, médiatement ou immediatement, d'un exposant plus petit que lui, j'apellerai cela un derangement. Qu'on compte, pour chaque terme, le nombre des dérangements: s'il est pair ou nul, le terme aura le signe +; s'il est impair, le terme aura le signe −. Par ex. dans le terme Z1Y2V3 il n'y a aucun dérangement: ce terme aura done le signe +. Le terme Z3Y1X2 a aussi le signe +, parce qu'il a deux derangements, 3 avant 1 & 3 avant 2. Mais le terme Z3Y2X1 qui a trois derangements, 3 avant 2, 3 avant 1, & 2 avant 1, aura le signe −.”

According to Cramer, therefore, If δ be the number of derangements in the permutation corresponding to any term, the sign of the term is (−)δ.

Instead of the word “dérangement,” Gergonne in 1813 used “inversion” in speaking of Cramer's rule: Cauchy did the same in 1841: and, consequently, the latter term or “inversion of order” has come into pretty general use. “Inverted-pair” is probably a still better expression.‡

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 22 , 1899 , pp. 441 - 477
Copyright
Copyright © Royal Society of Edinburgh 1899

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References

page 441 note * Annales de Math., iv. pp. 148–155Google Scholar.

page 441 note † Exercices d'analyse et de phys. math., ii. pp. 145–150Google Scholar.

page 441 note ‡ Proc. Roy. Soc. Edin., xvi. p. 449Google Scholar.

page 443 note * Journ. de l'Ec. Polyt., x. (Cah. xvii.) pp. 29–112Google Scholar.

page 443 note † Or below it. The arrow-head is useful in this connection.

page 463 note * “Two permutations of the numbers 1, 2, 3, …, n are called conjugate when each number and the number of the place which it occupies in the one permutation are interchanged in the case of the other permutation.” See Muir, , “History of Determinants,” pp. 59, 60Google Scholar; Muir, , “On Self-Conjugate Permutations.”—Proc. Roy. Soc. Edin., xvii. pp. 7–13Google Scholar.

page 467 note * This proposition is a modification of one of Rothe's. See p. 268 of his Memoir, or Muir's, “History of Determinants,” p. 56Google Scholar. Rothe's proof is very engthy.

page 473 note * See Muir, , “History of Determinants,” pages 91 …, 234 …, 259 …Google Scholar, or the papers themselves there referred to.

page 474 note * See Muir, , “On Self-Conjugate Permutations,” Proc. Roy. Soc. Edin., xvii. pp. 7–13Google Scholar.