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

CHAPTER VI.

COMPOUND DETERMINANTS, FROM 1862 TO 1880.
STILL greater progress than in the case of alternants has here to be
chronicled. Not only is the number of writings more than treble
the number for the preceding twenty-year period, but there is also
a large proportion of them of considerable importance.
Compound determinants of the special type associated with
Wronskians (Hist., ii. pp. 227-228) will be found referred to in
chap. viii.
ZEHFUSS, G. (1862).
[Zwei Satze iiber Determinanten. Zeitschrift f. Math. u. Phys.,
vii. pp. 436-439.]
Zehfuss' first theorem is one of the extreme cases of Bazin's of
the year 1851, namely,
I l 27.3.* -.n I I ala2 )/3 *. Xn a - - - | l2.* ~ I
I bl3273 *...* * Xn I (1b2 3 * * Xn I * * * a * * *. )n
I 1 273. * n 23 * I * l2Y | l3I2 * - * ' n I
= ab2... *1 I aL2*. An I. -
His proof is a lengthened way of saying that if we multiply the
compound determinant by I al3s273... n in row-by-column
fashion we obtain a determinant having for its elements the elements
of I ab2... 1,, 1, each multiplied by | a12y,... X,, and being
therefore equal to
I a1b..... I ~ 2-, ~ ~ ~ Xn
The theorem is used to prove Jacobi's theorem regarding a minor
of the adjugate.