Page 65 - Determinants and Their Applications in Mathematical Physics

50 3. Intermediate Determinant Theory

n n

= − v q y s A qs .
q=1 s=1
Similarly,

n n

B n+1,n+2 =+ u p y s A ps ,
p=1 s=1
n n

B n+2,n+1 =+ v q x r A qr ,
q=1 r=1
n n

B n+2,n+2 = − u p x r A pr .
p=1 r=1
Note the variations in the choice of dummy variables. Hence, (3.7.3)
becomes

n

BA = A pr A ps .
u p v q x r y s

p,q,r,s=1 A qr A qs
The theorem appears after applying the Jacobi identity and dividing
by A.
Exercises
1. Prove the Cauchy expansion formula for A ij , namely

n n

A ij = a pq A ip,jq − a ps a rq A ipr,jqs ,
r=1 s=1
where (p, q) =(i, j) but are otherwise arbitrary. Those terms in which
r = i or p or those in which s = j or q are zero by the definition of
higher cofactors.
2. Prove the generalized Cauchy expansion formula, namely

A = N ij,hk A ij,hk + N ij,rs N pq,hk A ijpq,rshk ,
1≤p≤q≤n 1≤r≤s≤n
where N ij,hk is a retainer minor and A ij,hk is its complementary
cofactor.