CHAPTER 23
SYSTEMS OF EQUATIONS, MATRICES AND DETERMINANTS

223. Introduction. The solution of systems of simultaneous linear equations is a problem which has been before mathematicians for centuries. It is known, for instance, that the Babylonians studied systems of up to 10 equations. The solution of large systems arose in connection with the adjustment of astronomical and geodetical observations and it was in this connection that Gauss made notable contributions. More recently, considerable attention has been paid to large systems which arise by the approximation to differential and integral equations and in studies in economics.

The study of such systems is linked up with the theory of determinants. Fourier, in the original investigations concerning the coefficients in a trigonometrical series representing a periodic analytic function solved such system (involving an arbitrary number of variables) without the use of determinants. The first systematic account of the theory of determinants is due to Cauchy. An axiomatic presentation is due to Weierstrass.

A most important aid to the study is the theory of matrices, which was introduced by Cayley in connection with the theory of transformations. Since then the theory of matrices has penetrated many branches of mathematics. Although matrices do not obey the commutative law of multiplication, they are of great service sometimes, in fact, for this very reason. They can be used to give representations of more abstract mathematical objects such as permutations, groups, hypercomplex systems. Thus the theory has, therefore, many applications in such branches of mathematics as modern algebra, geometry, number theory, the theory of automorphic functions.

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