Determinants and Their Applications in Mathematical Physics

3.3 The Laplace Expansion 29

The expansion formula (3.3.4) follows.

Illustrations

1.Whenr= 2, the Laplace expansion formula can be proved as follows:

Changing the notation in the second line of (3.3.10),

A=

n

∑

p=1

n

∑

q=1

aipajqAij;pq,i<j.

This double sum containsn

2

terms, but thenterms in whichq=pare

zero by the definition of a second cofactor. Hence,

A=

∑

p<q

aipajqAij,pq+

∑

q<p

aipajqAij;pq.

In the second double sum, interchange the dummiespandqand refer

once again to the definition of a second cofactor:

A=

∑

p<q

∣

∣

∣

∣

aip aiq

ajp ajq

∣

∣

∣

∣

Aij;pq

=

∑

p<q

Nij;pqAij;pq,i<j,

which proves the Laplace expansion formula from rowsiandj. When

(n, i, j)=(4, 1 ,2), this formula becomes

A=N 12 , 12 A 12 , 12 +N 12 , 13 A 12 , 13 +N 12 , 14 A 12 , 14

+N 12 , 23 A 12 , 23 +N 12 , 24 A 12 , 24

+N 12 , 34 A 12 , 34.

2.Whenr= 3, begin with the formula

A=

n

∑

p=1

n

∑

q=1

n

∑

r=1

aipajqakrAijk,pqr, i<j<k,

which is obtained from the second line of (3.3.11) with a change in

notation. The triple sum containsn

3

terms, but those in whichp,q,

andrare not distinct are zero. Those which remain can be divided into

3! = 6 groups according to the relative magnitudes ofp,q, andr:

A=

[

∑

p<q<r

+

∑

p<r<q

+

∑

q<r<p

+

∑

q<p<r

+

∑

r<p<q

+

∑

r<q<p

]

aipajqakrAijk,pqr.