Page 21 - Determinants and Their Applications in Mathematical Physics

6 1. Determinants, First Minors, and Cofactors

n

= c ij e j ,
j=1
where
n

c ij = a ik b kj . (1.4.2)
k=1
Hence,

x 1 x 2 ··· x n = |c ij | n e 1 e 2 ··· e n . (1.4.3)
Comparing (1.4.1) and (1.4.3),
|a ij | n |b ij | n = |c ij | n . (1.4.4)
Another proof of (1.4.4) is given in Section 3.3.5 by applying the Laplace
expansion in reverse.
The Laplace expansion formula is proved by both a Grassmann and a
classical method in Chapter 3 after the definitions of second and higher
rejector and retainor minors and cofactors.