Editing Gottfried Wilhelm Leibniz (section)

===Linear systems===
Leibniz arranged the coefficients of a system of [[linear equation]]s into an array, now called a [[Matrix (mathematics)|matrix]], in order to find a solution to the system if it existed.<ref>{{cite book|last1=Jones|first1=Matthew L.|title=The Good Life in the Scientific Revolution: Descartes, Pascal, Leibniz, and the Cultivation of Virtue|publisher=University of Chicago Press|isbn=978-0-226-40955-9|pages=237–239|date=2006-10-01}}</ref> This method was later called [[Gaussian elimination]]. Leibniz laid down the foundations and theory of [[determinants]], although the Japanese mathematician [[Seki Takakazu]] also discovered determinants independently of Leibniz.<ref>{{cite book|last1=Agarwal|first1=Ravi P|last2=Sen|first2=Syamal K|title=Creators of Mathematical and Computational Sciences|date=2014|publisher=Springer, Cham|isbn=978-3-319-10870-4|page=180}}</ref><ref name="Princeton University Press">{{cite book|editor-first1=Timothy|editor-last1=Gowers|editor-first2=June|editor-last2=Barrow-Green|editor-first3=Imre|editor-last3=Leader|title=The Princeton Companion to Mathematics|url=https://archive.org/details/princetoncompanio00gowe|date=2008|publisher=Princeton University Press|location=Princeton|isbn=978-0-691-11880-2|page=744}}</ref> His works show calculating the determinants using cofactors.<ref>{{cite book|last1=Knobloch|first1=Eberhard|authorlink=Eberhard Knobloch|title=Leibniz's Theory of Elimination and Determinants|date=13 March 2013|publisher=Springer|isbn=978-4-431-54272-8|pages=230–237}}</ref> Calculating the determinant using cofactors is named the [[Leibniz formula for determinants|Leibniz formula]]. Finding the determinant of a matrix using this method proves impractical with large ''n'', requiring to calculate ''n!'' products and the number of n-permutations.<ref>{{cite book|title=Concise Dictionary of Mathematics|publisher=V&S Publishers|isbn=978-93-81588-83-3|pages=113–114|date=April 2012}}</ref> He also solved systems of linear equations using determinants, which is now called [[Cramer's rule]]. This method for solving systems of linear equations based on determinants was found in 1684 by Leibniz ([[Gabriel Cramer]] published his findings in 1750).<ref name="Princeton University Press"/> Although Gaussian elimination requires <math>O(n^3)</math> arithmetic operations, linear algebra textbooks still teach cofactor expansion before [[LU factorization]].<ref>{{cite book|last1=Lay|first1=David C.|title=Linear algebra and its applications|date=2012|publisher=Addison-Wesley|location=Boston|isbn=978-0-321-38517-8|edition=4th}}</ref><ref>{{cite book|first1=Takeshi|last1=Tokuyama|display-authors=etal|title=Algorithms and Computation: 18th International Symposium, ISAAC 2007, Sendai, Japan, December 17–19, 2007 : proceedings|url=https://archive.org/details/algorithmscomput00toku|url-access=limited|date=2007|publisher=Springer|location=Berlin [etc.]|isbn=978-3-540-77120-3|page=[https://archive.org/details/algorithmscomput00toku/page/n613 599]}}</ref>