Determinants and Their Applications in Mathematical Physics

26 3. Intermediate Determinant Theory

Recalling the definitions of rejecter minorsM, retainer minorsN, and

cofactorsA, each with row and column parameters, it is found that

yi

1

···yi

r

=Ni

1 ...ir;j 1 ...jr

(

ej

1

···ej

r

)

,

z 1

∗

···zn=Mi 1 ...ir;j 1 ...jr

(

e 1

∗

···en

)

,

where, in this case, the symbol∗denotes that those vectors with suffixes

j 1 ,j 2 ,...,jrare omitted. Hence,

x 1 ···xn

=

∑

i 1 ...ir

(−1)

p

Ni 1 i 2 ...ir;j 1 j 2 ...jrMi 1 i 2 ...,ir;j 1 j 2 ...jr

(

ej 1 ···ejr

)(

e 1

∗

···en

)

.

By applying in reverse order the sequence of interchanges used to obtain

(3.3.2), it is found that

(

ej 1 ···ejr

)(

e 1

∗

···en

)

=(−1)

q

(e 1 ···en),

where

q=

n

∑

s=1

js−

1

2

r(r+1).

Hence,

x 1 ···xn=

[

∑

i 1 ...ir

(−1)

p+q

Ni 1 i 2 ...ir;j 1 j 2 ...jrMi 1 i 2 ...ir;j 1 j 2 ...jr

]

e 1 ···en

=

[

∑

i 1 ...ir

Ni

1 i 2 ...ir;j 1 j 2 ...jr

Ai

1 i 2 ...ir;j 1 j 2 ...jr

]

e 1 ···en.

Comparing this formula with (1.2.5) in the section on the definition of a

determinant, it is seen that

An=|aij|n=

∑

i 1 ...ir

Ni 1 i 2 ...ir;j 1 j 2 ...jrAi 1 i 2 ...ir;j 1 j 2 ...jr, (3.3.4)

which is the general form of the Laplace expansion ofAnin which the sum

extends over the row parameters. By a similar argument, it can be shown

thatAnis also equal to the same expression in which the sum extends over

the column parameters.

Whenr= 1, the Laplace expansion degenerates into a simple expansion

by elements from columnjor rowiand their first cofactors:

An=

∑

iorj

NijAij,

=

∑

iorj

aijAij.