Determinants and Their Applications in Mathematical Physics

6 1. Determinants, First Minors, and Cofactors

=

n

∑

j=1

cijej,

where

cij=

n

∑

k=1

aikbkj. (1.4.2)

Hence,

x 1 x 2 ···xn=|cij|ne 1 e 2 ···en. (1.4.3)

Comparing (1.4.1) and (1.4.3),

|aij|n|bij|n=|cij|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.