Editing Determinant (section)

== History ==
Historically, determinants were used long before matrices: A determinant was originally defined as a property of a [[system of linear equations]].
The determinant "determines" whether the system has a unique solution (which occurs precisely if the determinant is non-zero).
In this sense, determinants were first used in the Chinese mathematics textbook ''[[The Nine Chapters on the Mathematical Art]]'' (九章算術, Chinese scholars, around the 3rd century BCE). In Europe, solutions of linear systems of two equations were expressed by [[Gerolamo Cardano|Cardano]] in 1545 by a determinant-like entity.<ref>{{harvnb|Grattan-Guinness|2003|loc=§6.6}}</ref>

Determinants proper originated separately from the work of [[Seki Takakazu]] in 1683 in Japan and parallelly of [[Gottfried Leibniz|Leibniz]] in 1693.<ref>Cajori, F. [https://archive.org/details/ahistorymathema02cajogoog/page/n94 ''A History of Mathematics'' p.&nbsp;80]</ref><ref name="Campbell" /><ref>{{harvnb|Eves|1990|p=405}}</ref><ref>A Brief History of Linear Algebra and Matrix Theory at: {{cite web |url=http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html |title=A Brief History of Linear Algebra and Matrix Theory |access-date=2012-01-24 |url-status=dead |archive-url=https://web.archive.org/web/20120910034016/http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html |archive-date=2012-09-10 |df=dmy-all}}</ref> {{harvtxt|Cramer|1750}} stated, without proof, Cramer's rule.<ref>{{harvnb|Kleiner|2007|p=80}}</ref> Both Cramer and also {{harvtxt|Bézout|1779}} were led to determinants by the question of [[plane curve]]s passing through a given set of points.<ref>{{harvtxt|Bourbaki|1994|p=59}}</ref>

[[Vandermonde]] (1771) first recognized determinants as independent functions.<ref name="Campbell">Campbell, H: "Linear Algebra With Applications", pages 111–112. Appleton Century Crofts, 1971</ref> {{harvtxt|Laplace|1772}} gave the general method of expanding a determinant in terms of its complementary [[minor (matrix)|minors]]: Vandermonde had already given a special case.<ref>Muir, Sir Thomas, ''The Theory of Determinants in the historical Order of Development'' [London, England: Macmillan and Co., Ltd., 1906]. {{JFM|37.0181.02}}</ref> Immediately following, [[Joseph Louis Lagrange|Lagrange]] (1773) treated determinants of the second and third order and applied it to questions of [[elimination theory]]; he proved many special cases of general identities.

[[Carl Friedrich Gauss|Gauss]] (1801) made the next advance. Like Lagrange, he made much use of determinants in the [[theory of numbers]]. He introduced the word "determinant" (Laplace had used "resultant"), though not in the present signification, but rather as applied to the [[discriminant]] of a [[quadratic form]].<ref>{{harvnb|Kleiner|2007|loc=§5.2}}</ref> Gauss also arrived at the notion of reciprocal (inverse) determinants, and came very near the multiplication theorem.{{Clarify|date=June 2023|reason=What is "the multiplication theorem"?}}

The next contributor of importance is [[Jacques Philippe Marie Binet|Binet]] (1811, 1812), who formally stated the theorem relating to the product of two matrices of ''m'' columns and ''n'' rows, which for the special case of {{math|1=''m'' = ''n''}} reduces to the multiplication theorem. On the same day (November 30, 1812) that Binet presented his paper to the Academy, [[Cauchy]] also presented one on the subject. (See [[Cauchy–Binet formula]].) In this he used the word "determinant" in its present sense,<ref>The first use of the word "determinant" in the modern sense appeared in: Cauchy, Augustin-Louis "Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et des signes contraires par suite des transpositions operées entre les variables qu'elles renferment," which was first read at the Institute de France in Paris on November 30, 1812, and which was subsequently published in the ''Journal de l'Ecole Polytechnique'', Cahier 17, Tome 10, pages 29–112 (1815).</ref><ref>Origins of mathematical terms: http://jeff560.tripod.com/d.html</ref> summarized and simplified what was then known on the subject, improved the notation, and gave the multiplication theorem with a proof more satisfactory than Binet's.<ref name="Campbell" /><ref>History of matrices and determinants: http://www-history.mcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants.html</ref> With him begins the theory in its generality.

{{harvtxt|Jacobi|1841}} used the functional determinant which Sylvester later called the [[Jacobian matrix and determinant|Jacobian]].<ref>{{harvnb|Eves|1990|p=494}}</ref> In his memoirs in ''[[Crelle's Journal]]'' for 1841 he specially treats this subject, as well as the class of alternating functions which Sylvester has called ''alternants''. About the time of Jacobi's last memoirs, [[James Joseph Sylvester|Sylvester]] (1839) and [[Arthur Cayley|Cayley]] began their work. {{harvnb|Cayley|1841}} introduced the modern notation for the determinant using vertical bars.<ref>{{harvnb|Cajori|1993|loc=Vol. II, p. 92, no. 462}}</ref><ref>History of matrix notation: http://jeff560.tripod.com/matrices.html</ref>

The study of special forms of determinants has been the natural result of the completion of the general theory. Axisymmetric determinants have been studied by [[Lebesgue]], [[Otto Hesse|Hesse]], and Sylvester; [[persymmetric]] determinants by Sylvester and [[Hermann Hankel|Hankel]]; [[circulant]]s by [[Eugène Charles Catalan|Catalan]], [[William Spottiswoode|Spottiswoode]], [[James Whitbread Lee Glaisher|Glaisher]], and Scott; skew determinants and [[Pfaffian]]s, in connection with the theory of [[orthogonal transformation]], by Cayley; continuants by Sylvester; [[Wronskian]]s (so called by [[Thomas Muir (mathematician)|Muir]]) by [[Elwin Bruno Christoffel|Christoffel]] and [[Ferdinand Georg Frobenius|Frobenius]]; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and [[Hessian matrix|Hessians]] by Sylvester; and symmetric gauche determinants by [[Trudi]]. Of the textbooks on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.