26 3. Intermediate Determinant Theory
Recalling the definitions of rejecter minorsM, retainer minorsN, and
cofactorsA, each with row and column parameters, it is found that
yi
1
···yi
r
=Ni
1 ...ir;j 1 ...jr
(
ej
1
···ej
r
)
,
z 1
∗
···zn=Mi 1 ...ir;j 1 ...jr
(
e 1
∗
···en
)
,
where, in this case, the symbol∗denotes that those vectors with suffixes
j 1 ,j 2 ,...,jrare omitted. Hence,
x 1 ···xn
=
∑
i 1 ...ir
(−1)
p
Ni 1 i 2 ...ir;j 1 j 2 ...jrMi 1 i 2 ...,ir;j 1 j 2 ...jr
(
ej 1 ···ejr
)(
e 1
∗
···en
)
.
By applying in reverse order the sequence of interchanges used to obtain
(3.3.2), it is found that
(
ej 1 ···ejr
)(
e 1
∗
···en
)
=(−1)
q
(e 1 ···en),
where
q=
n
∑
s=1
js−
1
2
r(r+1).
Hence,
x 1 ···xn=
[
∑
i 1 ...ir
(−1)
p+q
Ni 1 i 2 ...ir;j 1 j 2 ...jrMi 1 i 2 ...ir;j 1 j 2 ...jr
]
e 1 ···en
=
[
∑
i 1 ...ir
Ni
1 i 2 ...ir;j 1 j 2 ...jr
Ai
1 i 2 ...ir;j 1 j 2 ...jr
]
e 1 ···en.
Comparing this formula with (1.2.5) in the section on the definition of a
determinant, it is seen that
An=|aij|n=
∑
i 1 ...ir
Ni 1 i 2 ...ir;j 1 j 2 ...jrAi 1 i 2 ...ir;j 1 j 2 ...jr, (3.3.4)
which is the general form of the Laplace expansion ofAnin which the sum
extends over the row parameters. By a similar argument, it can be shown
thatAnis also equal to the same expression in which the sum extends over
the column parameters.
Whenr= 1, the Laplace expansion degenerates into a simple expansion
by elements from columnjor rowiand their first cofactors:
An=
∑
iorj
NijAij,
=
∑
iorj
aijAij.