Observe that for a determinant of the n-th order, taking the decomposition to be 1 + (n - 1), we fall back upon the equations given at the commencement, in order to show the genesis of a determinant.
| 8. Any determinant | a, | b | formed out of the elements of the original determinant, by selecting the | ||
| a′, | b′ |
lines and columns at pleasure, is termed a minor of the original determinant; and when the number of lines and columns, or order of the determinant, is n-1, then such determinant is called a first minor; the number of the first minors is = n², the first minors, in fact, corresponding to the several elements of the determinant—that is, the coefficient therein of any term whatever is the corresponding first minor. The first minors, each divided by the determinant itself, form a system of elements inverse to the elements of the determinant.
A determinant is symmetrical when every two elements symmetrically situated in regard to the dexter diagonal are equal to each other; if they are equal and opposite (that is, if the sum of the two elements be = 0), this relation not extending to the diagonal elements themselves, which remain arbitrary, then the determinant is skew; but if the relation does extend to the diagonal terms (that is, if these are each = 0), then the determinant is skew symmetrical; thus the determinants
| a, | h, | g | ; | a, | ν, | -μ | ; | 0, | ν, | -μ | ||||||
| h, | b, | f | -ν, | b, | λ | -ν, | 0, | λ | ||||||||
| g, | f, | c | μ, | -λ, | c | μ, | -λ, | 0 |
are respectively symmetrical, skew and skew symmetrical: The theory admits of very extensive algebraic developments, and applications in algebraical geometry and other parts of mathematics. For further developments of the theory of determinants see [Algebraic Forms].
(A. Ca.)
9. History.—These functions were originally known as "resultants," a name applied to them by Pierre Simon Laplace, but now replaced by the title "determinants," a name first applied to certain forms of them by Carl Friedrich Gauss. The germ of the theory of determinants is to be found in the writings of Gottfried Wilhelm Leibnitz (1693), who incidentally discovered certain properties when reducing the eliminant of a system of linear equations. Gabriel Cramer, in a note to his Analyse des lignes courbes algébriques (1750), gave the rule which establishes the sign of a product as plus or minus according as the number of displacements from the typical form has been even or odd. Determinants were also employed by Étienne Bezout in 1764, but the first connected account of these functions was published in 1772 by Charles Auguste Vandermonde. Laplace developed a theorem of Vandermonde for the expansion of a determinant, and in 1773 Joseph Louis Lagrange, in his memoir on Pyramids, used determinants of the third order, and proved that the square of a determinant was also a determinant. Although he obtained results now identified with determinants, Lagrange did not discuss these functions systematically. In 1801 Gauss published his Disquisitiones arithmeticae, which, although written in an obscure form, gave a new impetus to investigations on this and kindred subjects. To Gauss is due the establishment of the important theorem, that the product of two determinants both of the second and third orders is a determinant. The formulation of the general theory is due to Augustin Louis Cauchy, whose work was the forerunner of the brilliant discoveries made in the following decades by Hoëné-Wronski and J. Binet in France, Carl Gustav Jacobi in Germany, and James Joseph Sylvester and Arthur Cayley in England. Jacobi's researches were published in Crelle's Journal (1826-1841). In these papers the subject was recast and enriched by new and important theorems, through which the name of Jacobi is indissolubly associated with this branch of science. The far-reaching discoveries of Sylvester and Cayley rank as one of the most important developments of pure mathematics. Numerous new fields were opened up, and have been diligently explored by many mathematicians. Skew-determinants were studied by Cayley; axisymmetric-determinants by Jacobi, V. A. Lebesque, Sylvester and O. Hesse, and centro-symmetric determinants by W. R. F. Scott and G. Zehfuss. Continuants have been discussed by Sylvester; alternants by Cauchy, Jacobi, N. Trudi, H. Nagelbach and G. Garbieri; circulants by E. Catalan, W. Spottiswoode and J. W. L. Glaisher, and Wronskians by E. B. Christoffel and G. Frobenius. Determinants composed of binomial coefficients have been studied by V. von Zeipel; the expression of definite integrals as determinants by A. Tissot and A. Enneper, and the expression of continued fractions as determinants by Jacobi, V. Nachreiner, S. Günther and E. Fürstenau. (See T. Muir, Theory of Determinants, 1906).
[1] The expression, a linear function, is here used in its narrowest sense, a linear function without constant term; what is meant is that the determinant is in regard to the elements a, a′, a″, ... of any column or line thereof, a function of the form Aa + A′a′ + A″a″ + ... without any term independent of a, a′, a″ ...