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

290 HISTORY OF THE THEORY OF DETERMINANTS

But alrAls + a2A2s+... +a.Ars +... + aAns = A when s = r
= 0 when s rJ,
and thence by addition, whatever s may be,
tir(As +As1) +.. + a,(As,+ Ans) = 2zAsr.
Writing Wsr for (As+ A,.) 2z he thus has the set of n equations
in W081, 0s2, ~ ~ ~, sn,
al(81 +.... + a(1n50sn = Als
a210(81 +.... + a2n )sn = A2s
anWsi8 +.... nsn = A s
the peculiarity of which is that the right-hand members are
cofactors of a column of elements of the determinant formed
from the coefficients on the left. The solution thus is.AiAl. + A2sA2r+.... + AnsAnr
08r = --
whence
(Al8, A28,s. AXIS A *r A2.... An, sr) A+rs A,
as desired. Using this n2 times, he, of course, obtains for the
square of the adjugate the expression
c011 C5012 ~ ~ ~ e0l1
21 1..21 ~ ~ ~ 2n
A... ~.
)cnl C)n2*... (Nn
and, it being known otherwise that the square of the adjugate is
A2W'-2, it follows that
W(11 W22.. * * n =- A-2,
which is the other result wanted.
In regard to the elements w one fact is noted, and is worth
noting. Since A,. may be expressed as an aggregate of terms in
z~, z1, z2,..., namely, say
Asr = 0+ e +Z 02 +....