History
Historically, determinants were used long before matrices: originally, a determinant was defined as a property of a
system of linear equations. The determinant "determines" whether the system has a unique solution (which occurs precisely if the determinant is non-zero). In this sense, determinants were first used in the Chinese mathematics textbook
The Nine Chapters on the Mathematical Art (九章算術, Chinese scholars, around the 3rd century BCE). In Europe, 2 × 2 determinants were considered by
Cardano at the end of the 16th century and larger ones by
Leibniz.
[19][20][21][22]
In Japan,
Seki Takakazu (関 孝和) is credited with the discovery of the resultant and the determinant (at first in 1683, the complete version no later than 1710). In Europe,
Cramer (1750) added to the theory, treating the subject in relation to sets of equations. The recurrence law was first announced by
Bézout (1764).
It was
Vandermonde (1771) who first recognized determinants as independent functions.
[19] Laplace (1772)
[23][24] gave the general method of expanding a determinant in terms of its complementary
minors: Vandermonde had already given a special case. Immediately following,
Lagrange (1773) treated determinants of the second and third order and applied it to questions of
elimination theory; he proved many special cases of general identities.
Gauss (1801) made the next advance. Like Lagrange, he made much use of determinants in the
theory of numbers. He introduced the word
determinant (Laplace had used
resultant), though not in the present signification, but rather as applied to the
discriminant of a
quantic. Gauss also arrived at the notion of reciprocal (inverse) determinants, and came very near the multiplication theorem.
The next contributor of importance is
Binet (1811, 1812), who formally stated the theorem relating to the product of two matrices of
m columns and
n rows, which for the special case of
m =
n reduces to the multiplication theorem. On the same day (November 30, 1812) that Binet presented his paper to the Academy,
Cauchy also presented one on the subject. (See
Cauchy–Binet formula.) In this he used the word
determinant in its present sense,
[25][26] summarized and simplified what was then known on the subject, improved the notation, and gave the multiplication theorem with a proof more satisfactory than Binet's.
[19][27] With him begins the theory in its generality.
The next important figure was
Jacobi[20] (from 1827). He early used the functional determinant which Sylvester later called the
Jacobian, and in his memoirs in
Crelle's Journal for 1841 he specially treats this subject, as well as the class of alternating functions which Sylvester has called
alternants. About the time of Jacobi's last memoirs,
Sylvester (1839) and
Cayley began their work.
[28][29]
The study of special forms of determinants has been the natural result of the completion of the general theory. Axisymmetric determinants have been studied by
Lebesgue,
Hesse, and Sylvester;
persymmetric determinants by Sylvester and
Hankel;
circulants by
Catalan,
Spottiswoode,
Glaisher, and Scott; skew determinants and
Pfaffians, in connection with the theory of
orthogonal transformation, by Cayley; continuants by Sylvester;
Wronskians (so called by
Muir) by
Christoffel and
Frobenius; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and
Hessians by Sylvester; and symmetric gauche determinants by
Trudi. Of the textbooks on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.
