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

DETERMINANTS IN GENERAL (CAYLEY, 1860)

5

The two agree as far as the third line only: the fourth and fifth
lines above are combined into one in the earlier development, the
cofactor of 151 appearing there as a zero-axial determinant; and a
similar substitute appears for the aggregate of the sixth and seventh
lines.
SMITH, H. J. S. (1861).
[On systems of linear indeterminate equations and congruences.
Transac. R. Soc. London, cli. pp. 293-326; or Collected Math.
Papers, i. pp. 367-409.]
As the equations dealt with are for solution in integral numbers,
the subject of this memoir belongs strictly to the Theory of Integers.
There are, however, several points of contact with Determinants,
just as in the case of previous papers of the same kind by Hermite
(1849), Bazin (1854), Heger (1856).
Thus, he resuscitates Cayley's row-by-column multiplication of
an n-line determinant by an array of n rows and n+h columns
(1843, Hist., ii. p. 16), using, for example, the statement
a1 a2 c| C2 C3 C4 I _ x1 x2 X3 X4
b, b2 do d2 d3 d4 i- Yx Y2 Y3 Y4
to stand for six equalities of the type
I ab21 -I cd2 = I XlY 2
In this connection, too, he draws attention to a necessary distinction by saying that j a1b2 I is here postmultiplied * by I cd2 1,
and that I c1d2 1 is premultiplied by I ab2 i.
The main subject of the memoir was continued in three papers
-published under the conjoint title "Arithmetical Notes" in the
Proceed. London Math. Soc., iv. (.1873), pp. 236-253.
* In Cayley's other theorem of 1843 the determinant is premultiplied by an
array, for example,
C1 C2,C C4 \1 \ \3 X4 4
I do dc13 d4 A1' / L2 3 /4 1?72 973 774
Yl V2 Y,3 4
Pi P2 P3 P4
which stands for six equalities of the type
II cd4[ * I X 41 = I v2 1,
the two factors on the left being now, not determinants, but 2-by-4 arrays.