224 V Hadamard’s Determinant Problem
has, ifδ 2 =α 11 α 22 −α 12 α 21 is nonzero, the unique solution
ξ 1 =(β 1 α 22 −β 2 α 12 )/δ 2 ,ξ 2 =−(β 1 α 21 −β 2 α 11 )/δ 2.Ifδ 2 =0, then either there is no solution or there is more than one solution.
Similarly the system of three simultaneous linear equations
α 11 ξ 1 +α 12 ξ 2 +α 13 ξ 3 =β 1
α 21 ξ 1 +α 22 ξ 2 +α 23 ξ 3 =β 2
α 31 ξ 1 +α 32 ξ 2 +α 33 ξ 3 =β 3has a unique solution if and only ifδ 3 =0, where
δ 3 =α 11 α 22 α 33 +α 12 α 23 α 31 +α 13 α 21 α 32
−α 11 α 23 α 32 −α 12 α 21 α 33 −α 13 α 22 α 31.These considerations may be extended to any finite number of simultaneous linear
equations. The system
α 11 ξ 1 +α 12 ξ 2 +···+α 1 nξn=β 1
α 21 ξ 1 +α 22 ξ 2 +···+α 2 nξn=β 2
···
αn 1 ξ 1 +αn 2 ξ 2 +···+αnnξn=βnhas a unique solution if and only ifδn=0, where
δn=∑
±α 1 k 1 α 2 k 2 ···αnkn,the sum being taken over alln! permutationsk 1 ,k 2 ,...,knof 1, 2 ,...,nand the sign
chosen being+or−according as the permutation is even or odd, as defined in Chap-
ter I,§7.
It has been tacitly assumed that the given quantitiesαjk,βj(j,k= 1 ,...,n)are
real numbers, in which case the solutionξk(k= 1 ,...,n)also consists of real num-
bers. However, everything that has been said remains valid if the given quantities are
elements of an arbitrary fieldF, in which case the solution also consists of elements
ofF.Sinceδnis an element ofFwhich is uniquely determined by the matrix
A=
⎡
⎣
α 11 ··· α 1 n
···
αn 1 ··· αnn⎤
⎦,
it will be called thedeterminantof the matrixAand denoted by detA.
Determinants appear in the work of the Japanese mathematician Seki (1683) and
in a letter of Leibniz (1693) to l’Hospital, but neither had any influence on later
developments. The rule which expresses the solution of a system of linear equations by
quotients of determinants was stated by Cramer (1750), but the study of determinants
for their own sake began with Vandermonde (1771). The word ‘determinant’ was first
used in the present sense by Cauchy (1812), who gave a systematic account of their