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

DETERMINANTS IN GENERAL (JANNI, 1874)

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JANNI, V. (1874).
[Dimostrazione di alcune teoremi sui determinanti. Giornale di
Mat., xii. pp. 142-145.]
Janni's theorems, though all simple, are treated with freshness.
His first is that dealt with very shortly by Bellavitis in ~ 15 of his
Sposizione of 1857 (Hist., ii. pp. 94-95), his proof resting on the
fact that, for example, when acb2c3 = 0, we not only have
blAl+b2A2+b3A3 = 0,
and c1A1+c2A2+C3A5 = 0,
but also aAl1+a2A2+a3A3 = 0.
Following on this are other properties of a null determinant
established in his own way. He then proves Hermite's condensation-theorem (Hist., ii. p. 46), deducing thence the corollary that
if all the two-line minors of a determinant be divisible by the same
quantity, any m-line minor will be divisible by the (m —)th power
of the quantity, and deriving therefrom the two main properties
of the adjugate. His last result is that a determinant is not altered
by changing the sign of every element whose place-indices have an odd
sum, this change being equivalent to changing the signs of the
even-numbered rows and thereafter the signs of the even-numbered
columns.
KRONECKER, L. (1874).
[Ueber Schaaren von quadratischen und bilinearen Formen.
Monatsb.... Akad. d. Wiss. (Berlin), Jahrg. 1874, pp....,
214-215,...; or Werke, i. pp. 165-174.]
Using the simplest case of Jacobi's theorem regarding a minor
of the adjugate Kronecker makes a deduction, of which it will
suffice to give merely an example, namely,
I abc3d, 4 Ij a c3c I -bc,,dl + I I a,41* I _ 1_, +_ d _,
+- 4 + - di +,
l c3d4l I b2c3d41 * C3d I C3d41.4 4
the reader being referred to Hist., ii. p. 40, and to a result given
under Axisymmetric Determinants (1874).