Page 35 - Determinants and Their Applications in Mathematical Physics

20 3. Intermediate Determinant Theory

In the definition of rejecter and retainer minors, no restriction is made
concerning the relative magnitudes of either the row parameters i s or the
column parameters j s . Now, let each set of parameters be arranged in
ascending order of magnitude, that is,

i s <i s+1 ,j s <j s+1 , 1 ≤ s ≤ r − 1.
(n)
Then, the rth cofactor of A n , denoted by A is defined as a
i 1 i 2 ...i r ;j 1 j 2 ...j r
signed rth rejecter minor:
(n) (n)
A =(−1) M , (3.2.2)
k
i 1 i 2 ...i r ;j 1 j 2 ...j r i 1 i 2 ...i r ;j 1 j 2 ...j r
where k is the sum of the parameters:
r

k = (i s + j s ).
s=1
However, the concept of a cofactor is more general than that of a signed
minor. The definition can be extended to zero values and to all positive and
negative integer values of the parameters by adopting two conventions:

i. The cofactor changes sign when any two row parameters or any two
column parameters are interchanged. It follows without further assump-
tions that the cofactor is zero when either the row parameters or the
column parameters are not distinct.
ii. The cofactor is zero when any row or column parameter is less than 1
or greater than n.
Illustration.
(4) (4) (4) (4) (4)
A 12,23 = −A 21,23 = −A 12,32 = A 21,32 = M 12,23 = N 34,14 ,
(6) (6) (6) (6) (6)
A = −A = A = A = −M = −N 246,146 ,
135,235 135,253 135,523 315,253 135,235
(n) (n) (n)
A i 2 i 1 i 3 ;j 1 j 2 j 3 = −A i 1 i 2 i 3 ;j 1 j 2 j 3 = A i 1 i 2 i 3 ;j 1 j 3 j 2 ,
(n)
A =0 if p< 0
i 1 i 2 i 3 ;j 1 j 2 (n−p)
or p ≥ n
or p = n − j 1
or p = n − j 2 .

3.2.3 The Expansion of Cofactors in Terms of Higher
Cofactors
(n)
Since the first cofactor A is itself a determinant of order (n − 1), it can
ip
be expanded by the (n−1) elements from any row or column and their first
(n)
cofactors. But, first, cofactors of A are second cofactors of A n . Hence, it
ip