The theory of determinants in the historical order of development, by Sir Thomas Muir.

308 HISTORY OF THE THEORY OF DETERMINANTS

Lagrange's determinantal equation shall have all its roots
positive. The result is not so interesting in connection with our
present subject as a theorem made use of in the process of
attaining it, namely:-If we have given the set of equations
a11X1 + a12X2 +.. + ainxn = 0^x
a21X1 + ao92X2 +.. ~ 1 n = X2
anlx + an2x2 + + arx, = Oxn
where ars = asr, and if we put A for 1 ala..22 a. nn, then
A11 A21 + +... + Anx,, = x
A12x1 + A22x2 +... + AA2xn = A
Alx, + A2nX2 +...+ Anx o n
No proof of this is given, but one is readily got by using the
elements of the rth column of the adjugate of A as multipliers in
connection with the given set of equations and performing
addition, when there results the rth equation of the required set.*
*It should be noted that the theorem holds when A is any determinant whatever. Further, there is implied in it another of at least equal importance,
namely:-If A stand for I a1a22.. a..n, the equation whose roots are A times the
reciprocals of the roots of the equation
a,1 - 0 a12... aI
a21 a22-0 ~. a 2 =
al a,2.. a, nn-
is
An - 0 A2... An
A21 A2 - 0... A2g 0.
A,, An2... A1-e0.
An independent proof of this is readily obtained by substituting A/0 for 0 in the
original equation, expanding the determinant in a series arranged according to
descending powers of A/0, using 0"/A as a multiplier, substituting Al,, A12.
for their equivalents, and returning to the determinant form.