Page 37 - Determinants and Their Applications in Mathematical Physics

22 3. Intermediate Determinant Theory

The (n − r) values of p for which the expansion is valid correspond to the
(n − r) possible ways of expanding a subdeterminant of order (n − r)by
elements from one row and their cofactors.
If one of the column parameters of an rth cofactor of A n+1 is (n + 1),
the cofactor does not contain the element a n+1,n+1 . If none of the row
parameters is (n + 1), then the rth cofactor can be expanded by elements
from its last row and their first cofactors. But first cofactors of an rth
cofactor of A n+1 are (r + 1)th cofactors of A n+1 which, in this case, are
rth cofactors of A n . Hence, in this case, an rth cofactor of A n+1 can be
expanded in terms of the first n elements in the last row and rth cofactors
of A n . This expansion is
n
(n+1) (n)
A = − a n+1,q A . (3.2.8)
i 1 i 2 ...i r ;j 1 j 2 ...j r−1 (n+1) i 1 i 2 ...i r ;j 1 j 2 ...j r−1 q
q=1
The corresponding column expansion is

n
(n+1) (n)
A = − a p,n+1 A . (3.2.9)
i 1 i 2 ...i r−1 (n+1);j 1 j 2 ...j r i 1 i 2 ...i r−1 p;j 1 j 2 ...j r
p=1
Exercise. Prove that
2
2
∂ A ∂ A
= − ∂a iq ∂a jp ,
∂a ip ∂a jq
3
3
3
∂ A ∂ A ∂ A
= ∂a kp ∂a iq ∂a jr = ∂a jp ∂a kq ∂a ir
∂a ip ∂a jq ∂a kr
without restrictions on the relative magnitudes of the parameters.
3.2.4 Alien Second and Higher Cofactors; Sum Formulas

The (n − 2) elements a hq ,1 ≤ q ≤ n, q = h or p, appear in the second
(n)
cofactor A if h = i or j. Hence,
ij,pq
n
(n)
a hq A =0, h = i or j,
ij,pq
q=1
since the sum represents a determinant of order (n − 1) with two identical
rows. This formula is a generalization of the theorem on alien cofactors
given in Chapter 2. The value of the sum of 1 ≤ h ≤ n is given by the sum
formula for elements and cofactors, namely


(n)
 A , h = j = i

n
(n) ip
a hq A = (n) (3.2.10)
ij,pq −A ,h = i = j
 jp
q=1  0, otherwise