TITLE: What is the origin of the determinant in linear algebra?
QUESTION [48 upvotes]: We often learn in a standard linear algebra course that a determinant is a number associated with a square matrix. We can define the determinant also by saying that it is the sum of all the possible configurations picking an element from a matrix from different rows and different columns multiplied by (-1) or (1) according to the number inversions. 
But how is this notion of a 'determinant' derived? What is a determinant, actually? I searched up the history of the determinant and it looks like it predates matrices. How did the modern definition of a determinant come about? Why do we need to multiply some terms of the determinant sum by (-1) based on the number of inversions? I just can't understand the motivation that created determinants. We can define determinants, and see their properties, but I want to understand how they were defined and why they were defined to get a better idea of their important and application.

REPLY [17 votes]: The determinant was originally `discovered' by Cramer when solving systems of linear equations necessary to determine the coefficients of a polynomial curve passing through a given set of points. Cramer's rule, for giving the general solution of a system of linear equations, was a direct result of this.
This appears in Gabriel Cramer, ``Introduction a l'analyse des lignes courbes algebriques,''(Introduction to the analysis of algebraic line curves), Geneve, Ches les Freres Cramer & Cl. Philibert, (1750). It is cited as a footnote on p. 60, which reads (from French):
``I think I have found [for solving these equations] a very simple and general rule, when the number of equations and unknowns do not pass the first degree [e.g. are linear]. One finds this in the Appendix No. 1.'' Appendix No. 1 appears on p. 657 of the same text. The text is available on line, for those who can read French.
The history of the Determinant appears in Thomas Muir, ``The Theory of Determinants in the Historical Order of Development,'' Dover, NY, (1923). This is also available on line.