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In-Depth Information
n
det
(
A
)=
a
ij
∆
ij
,
j
=1
which is called
Laplace expansion
.
From this representation, if
det
(
A
)isnotequalto0,then
∆
ij
=0for
{
a
i
1
,a
i
2
,
···
,a
in
}
which are not equal to 0. Thus, the following proposition
is obtained.
Proposition 1.
If det
(
A
)
is not equal to 0 if at least one co-factor of a
ij
(
=0)
,
∆
ij
is not equal to 0.
It is notable that the above definition of a determinant gives the relation
between a original matrix
A
and submatrices (co-factors). Since cofactors gives
a square matrix of size
n
1, the above proposition gives the relation between
a matrix of size
n
and submatrices of size
n
−
1. In the same way, we can
discuss the relation between a corresponding matrix of size
n
and submatrices
of size
r
(1
−
≤
r<n
−
1).
Rank and Submatrix
Let us assume that corresponding matrix and submatrix are square (
n
×
n
and
r
×
r
, respectively).
Theorem 2.
If the rank of a corresponding matrix of size n
×
n is equal to r,
at least the determinant of one submatrix of size r
×
r is not equal to 0. That
is, there exists a submatrix A
i
1
i
2
···i
r
j
1
j
2
···j
r
, which satisfies det
(
A
i
1
i
2
···i
r
j
1
j
2
···j
r
)
=0
Corollary 1.
If the rank of a corresponding matrix of size n
n is equal to r,
all the determinants of the submatrices whose number of columns and rows
are larger than r
+1(
×
≤
n
)
are equal to 0.
3 Degree of Dependence
3.1 Determinantal Divisors
From the subdeterminants of all the submatrices of size 2, all the subdeter-
minants of a corresponding matrix has the greatest common divisor, equal
to 3.
From the recursive definition of the determinants, it is show that the sub-
determinants of size
r
+ 1 will have the greatest common divisor of the sub-
determinants of size
r
as a divisor. Thus,
Theorem 3.
Let d
k
(
A
)
denote the greatest common divisor of all the
subdeterminants of size k, det
(
A
i
1
i
2
···i
k
j
1
j
2
···j
r
,d
n
(
A
)
are called
determinantal divisors. From the definition of Laplace expansion,
)
. d
1
(
A
)
,d
2
(
A
)
,
···
d
k
(
A
)
|d
k
+1
(
A
)
.
