3.2 Second and Higher Minors and Cofactors 19
called arejecterminor. The numbersisandjsare known respectively as
row and column parameters.
Now, letNi
1 i 2 ...ir;j 1 j 2 ...jr
denote the subdeterminant of orderrwhich is
obtained fromAnbyretainingrowsi 1 ,i 2 ,...,irand columnsj 1 ,j 2 ,...,jr
and rejecting the other rows and columns.Ni 1 i 2 ...ir;j 1 j 2 ...jr may conve-
niently be called aretainerminor.
Examples.
M
(5)
13 , 25
=
∣ ∣ ∣ ∣ ∣ ∣
a 21 a 23 a 24
a 41 a 43 a 44
a 51 a 53 a 54
∣ ∣ ∣ ∣ ∣ ∣
=N 245 , 134 ,
M
(5)
245 , 134
=
∣
∣
∣
∣
a 12 a 15
a 32 a 35
∣
∣
∣
∣
=N 13 , 25.
The minorsM
(n)
i 1 i 2 ...ir;j 1 j 2 ...jr
andNi
1 i 2 ...ir;j 1 j 2 ...jr
are said to be mutually
complementary inAn, that is, each is the complement of the other inAn.
This relationship can be expressed in the form
M
(n)
i 1 i 2 ...ir;j 1 j 2 ...jr= compNi 1 i 2 ...ir;j 1 j 2 ...jr,
Ni
1 i 2 ...ir;j 1 j 2 ...jr
= compM
(n)
i 1 i 2 ...ir;j 1 j 2 ...jr
. (3.2.1)
The order and structure of rejecter minors depends on the value ofnbut
the order and structure of retainer minors are independent ofnprovided
only thatnis sufficiently large. For this reason, the parameternhas been
omitted fromN.
Examples.
Nip=
∣
∣a
ip
∣
∣
1
=aip,n≥ 1 ,
Nij,pq=
∣
∣
∣
∣
aip aiq
ajp ajq
∣
∣
∣
∣
,n≥ 2 ,
Nijk,pqr=
∣ ∣ ∣ ∣ ∣ ∣
aip aiq air
ajp ajq ajr
akp akq akr
∣ ∣ ∣ ∣ ∣ ∣
,n≥ 3.
Both rejecter and retainer minors arise in the construction of the Laplace
expansion of a determinant (Section 3.3).
Exercise.Prove that
∣
∣
∣
∣
Nij,pq Nij,pr
Nik,pq Nik,pr
∣
∣
∣
∣
=NipNijk,pqr.
3.2.2 Second and Higher Cofactors.............
The first cofactorA
(n)
ij
is defined in Chapter 1 and appears in Chapter 2.
It is now required to generalize that concept.