® For a square matrix A in matrix equation AX = B (i) |A| ≠ , there exists unique solution (ii) |A| = and ( adj A) B ≠ , then there exists no solution (iii) |A| = and ( adj A) B = , then system may or may not be consistent. DETERMINANTS Historical Note The Chinese method of representing the coefficients of the unknowns of several linear equations by using rods on a calculating board naturally led to the discovery of simple method of elimination. The arrangement of rods was precisely that of the numbers in a determinant. The Chinese, therefore, early developed the idea of subtracting columns and rows as in simplification of a determinant Mikami, China , pp , .
Seki Kowa, the greatest of the Japanese Mathematicians of seventeenth century in his work ‘ Kai Fukudai no Ho ’ in showed that he had the idea of determinants and of their expansion. But he used this device only in eliminating a quantity from two equations and not directly in the solution of a set of simultaneous linear equations. T. Hayashi, “ The Fakudoi and Determinants in Japanese Mathematics ,” in the proc.
of the Tokyo Math. Soc., V. Vendermonde was the first to recognise determinants as independent functions. He may be called the formal founder.
Laplace ( ), gave general method of expanding a determinant in terms of its complementary minors. In Lagrange treated determinants of the second and third orders and used them for purpose other than the solution of equations. In , Gauss used determinants in his theory of numbers. The next great contributor was Jacques - Philippe - Marie Binet, ( ) who 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.
Also on the same day, Cauchy ( ) presented one on the same subject. He used the word ‘determinant’ in its present sense. He gave the proof of multiplication theorem more satisfactory than Binet’s. The greatest contributor to the theory was Carl Gustav Jacob Jacobi, after this the word determinant received its final acceptance.