Determinants and Their Applications in Mathematical Physics

50 3. Intermediate Determinant Theory

=−

n

∑

q=1

n

∑

s=1

vqysAqs.

Similarly,

Bn+1,n+2=+

n

∑

p=1

n

∑

s=1

upysAps,

Bn+2,n+1=+

n

∑

q=1

n

∑

r=1

vqxrAqr,

Bn+2,n+2=−

n

∑

p=1

n

∑

r=1

upxrApr.

Note the variations in the choice of dummy variables. Hence, (3.7.3)

becomes

BA=

n

∑

p,q,r,s=1

upvqxrys

∣

∣

∣

∣

Apr Aps

Aqr Aqs

∣

∣

∣

∣

.

The theorem appears after applying the Jacobi identity and dividing

byA.

Exercises

1.Prove the Cauchy expansion formula forAij, namely

Aij=apqAip,jq−

n

∑

r=1

n

∑

s=1

apsarqAipr,jqs,

where (p, q)=(i, j) but are otherwise arbitrary. Those terms in which

r=iorpor those in whichs=jorqare zero by the definition of

higher cofactors.

2.Prove the generalized Cauchy expansion formula, namely

A=Nij,hkAij,hk+

∑

1 ≤p≤q≤n

∑

1 ≤r≤s≤n

Nij,rsNpq,hkAijpq,rshk,

where Nij,hk is a retainer minor and Aij,hk is its complementary

cofactor.