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

16 HISTORY OF THE THEORY OF DETERMINANTS
The introduction of the notation is fully justified by two
theorems which follow. The first is virtually to the effect that
we may multiply both sides of (3) by the determinant
X X 2...
(5) i t2 * * An
T1 T2 ~ Tn
just as if (3) were a single equation instead of Cn,q+l equations,
and as if the left-hand side were a determinant; and the result,
written in the form
XK A +...+XAnK /iKuA+...+/nAn,. A. T K +...+a O7
X.IKI + * + XnKn utK, +... + 1AnK91.. K1 +..- +r Tn
will be true; that is to say, we shall have a new set of Cn,q+
equations, which follows logically from the original set. Further.
and conversely, if the set (6) hold, we can deduce the set (3)
provided that the determiniiant (5) be not zero. The other
theorem is quite similar, being to the effect that the equations
(3) may be replaced by the set
x1 x2...
(8) A X1A +...... X1A+i...- +l + = 0,
XqA +... + qKi XqA2+.*. +oq... XA b +.. +wqKn
and that conversely from the set (8) the set (3) is deducible
provided the determinant..... 1
X2 q.. 2.
be not zero.
As the " derivation of coexistence " came prominently before us
in examining Sylvester's early work, it may be noted here in
passing that Cayley's second chapter, extending to about a page,
consists of the enunciation of a theorem on this subject.