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Origin of linear and determinant systems


In ancient western mathematics there are few appearances of systems of linear equations. In the East, however, the subject deserved much greater attention. With their special taste for diagrams, the Chinese represented linear systems by their coefficients written with bamboo bars on the squares of a board. So they eventually discovered the elimination resolution method - which consists of nullifying coefficients by elementary operations. Examples of this procedure are found in the Nine chapters on the art of mathematics, a text probably dating from the 111th century BC.

But it was not until 1683, in a work by the Japanese Seki Kowa, that the idea of ​​determinant (as a polynomial that is associated with a square of numbers) came to light. Kowa, considered the greatest Japanese mathematician of the seventeenth century, came to this notion through the study of linear systems, systematizing the old Chinese procedure (for two equations only).

The use of determinants in the West began ten years later in a work by Leibniz, also linked to linear systems. In short, Leibniz established the compatibility condition of a system of three equations with two unknowns in terms of the determinant of order 3 formed by the coefficients and the independent terms (this determinant must be null). To this end, he even created a notation with indexes for the coefficients: what today, for example, we would write as the12, Leibniz indicated by 12.

Cramer's well-known rule for solving systems of unknown unknown equations by determinants is in fact a discovery by the Scottish Colin Maclaurin (1698-1746), probably dating from 1729, though only published posthumously in 1748 in his Treatise of algebra. . But the name of the Swiss Gabriel Cramer (1704-1752) does not appear in this episode completely free of charge. Cramer also came to the rule (independently), but later, in his Introduction to Flat Curve Analysis (1750), in connection with the problem of determining the coefficients of the general conic A + By + Cx + Dy2 + Exy + x2 = 0.

The French Étienne Bézout (1730-1783), author of mathematical texts of success in his time, systematized in 1764 the process of establishing the signs of the terms of a determinant. And it was up to another Frenchman, Alexandre Vandermonde (1735-1796), in 1771 to undertake the first approach to determinant theory independent of the study of linear systems - though he also used them in solving these systems. Laplace's important theorem, which allows the expansion of a determinant through the shortest of selected rows and their respective algebraic complements, was demonstrated the following year by Laplace himself in an article which, judging by its title, had nothing to do with the subject. : "Research on Integral Calculus and the World System".

The determinant term, in its present meaning, appeared in 1812 in Cauchy's work on the subject. In this paper, presented to the Academy of Sciences, Cauchy summarized and simplified what was previously known about determinants, improved the notation (but the current one with two vertical bars flanking the square of numbers would only emerge in 1841 with Arthur Cayley) and gave a demonstration. of the determinant multiplication theorem - months earlier JFM Binet (1786-1856) had given the first demonstration of this theorem, but Cauchy's was superior.

In addition to Cauehy, the German contributor to the theory of determinants was Carl G. J. Jacobi (1804-1851), sometimes known as "the great algorithm." It is due to him the simple way in which this theory is presented today elementally. As an algorithm, Jacobi was a determinant notation enthusiast with its potentialities. Thus, the important Jacobean concept of a function, highlighting one of the most characteristic points of his work, is a tribute to the fairest.

HYGINO H. SUNDAYS* Submitted by user Jaime Batista

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