Page 34 - Determinants and Their Applications in Mathematical Physics

3.2 Second and Higher Minors and Cofactors 19

called a rejecter minor. The numbers i s and j s are known respectively as
row and column parameters.
denote the subdeterminant of order r which is
Now, let N i 1 i 2 ...i r ;j 1 j 2 ...j r
obtained from A n by retaining rows i 1 ,i 2 ,...,i r and columns j 1 ,j 2 ,...,j r
may conve-
and rejecting the other rows and columns. N i 1 i 2 ...i r ;j 1 j 2 ...j r
niently be called a retainer minor.
Examples.

a 21 a 23

(5) a 24
M 13,25 = a 41 a 43 = N 245,134 ,

a 44
a 51 a 53 a 54

(5) a 12
M 245,134 = a 15 = N 13,25 .
a 32 a 35

(n)
The minors M and N i 1 i 2 ...i r ;j 1 j 2 ...j r are said to be mutually
i 1 i 2 ...i r ;j 1 j 2 ...j r
complementary in A n , that is, each is the complement of the other in A n .
This relationship can be expressed in the form
(n)
M = comp N i 1 i 2 ...i r ;j 1 j 2 ...j r ,
i 1 i 2 ...i r ;j 1 j 2 ...j r
(n)
= comp M . (3.2.1)
N i 1 i 2 ...i r ;j 1 j 2 ...j r
i 1 i 2 ...i r ;j 1 j 2 ...j r
The order and structure of rejecter minors depends on the value of n but
the order and structure of retainer minors are independent of n provided
only that n is sufficiently large. For this reason, the parameter n has been
omitted from N.
Examples.

= a ip , n ≥ 1,

1
N ip = a ip

N ij,pq = a ip a iq , n ≥ 2,
a jp a jq

a ip a iq a ir
, n ≥ 3.
N ijk,pqr = a jp
a jq a jr
a kp a kq a kr
Both rejecter and retainer minors arise in the construction of the Laplace
expansion of a determinant (Section 3.3).
Exercise. Prove that

N ij,pq N ij,pr = N ip N ijk,pqr .
N ik,pq N ik,pr
3.2.2 Second and Higher Cofactors
(n)
The first cofactor A is defined in Chapter 1 and appears in Chapter 2.
ij
It is now required to generalize that concept.