Determinant
of a square matrix of order
over a commutative associative ring
with unit 1
The element of equal to the sum of all terms of the form
![]() |
where is a permutation of the numbers
and
is the number of inversions of the permutation
. The determinant of the matrix
![]() |
is written as
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The determinant of the matrix contains
terms; when
,
, when
,
. The most important instances in practice are those in which
is a field (especially a number field), a ring of functions (especially a ring of polynomials) or a ring of integers.
From now on, is a commutative associative ring with 1,
is the set of all square matrices of order
over
and
is the identity matrix over
. Let
, while
are the rows of the matrix
. (All that is said from here on is equally true for the columns of
.) The determinant of
can be considered as a function of its rows:
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The mapping
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is subject to the following three conditions:
1) is a linear function of any row of
:
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where ;
2) if the matrix is obtained from
by replacing a row
by a row
,
, then
;
3) .
Conditions 1)–3) uniquely define , i.e. if a mapping
satisfies conditions 1)–3), then
. An axiomatic construction of the theory of determinants is obtained in this way.
Let a mapping satisfy the condition:
) if
is obtained from
by multiplying one row by
, then
. Clearly 1) implies
). If
is a field, the conditions 1)–3) prove to be equivalent to the conditions
), 2), 3).
The determinant of a diagonal matrix is equal to the product of its diagonal entries. The surjectivity of the mapping follows from this. The determinant of a triangular matrix is also equal to the product of its diagonal entries. For a matrix
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where and
are square matrices,
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It follows from the properties of transposition that , where
denotes transposition. If the matrix
has two identical rows, its determinant equals zero; if two rows of a matrix
change places, then its determinant changes its sign;
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when ,
; for
and
from
,
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Thus, is an epimorphism of the multiplicative semi-groups
and
.
Let , let
be an
-matrix, let
be an
-matrix over
, and let
. Then the Binet–Cauchy formula holds:
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Let , and let
be the cofactor of the entry
. The following formulas are then true:
![]() | (1) |
where is the Kronecker symbol. Determinants are often calculated by development according to the elements of a row or column, i.e. by the formulas (1), by the Laplace theorem (see Cofactor) and by transformations of
which do not alter the determinant. For a matrix
from
, the inverse matrix
in
exists if and only if there is an element in
which is the inverse of
. Consequently, the mapping
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where is the group of all invertible matrices in
(i.e. the general linear group) and where
is the group of invertible elements in
, is an epimorphism of these groups.
A square matrix over a field is invertible if and only if its determinant is not zero. The -dimensional vectors
over a field
are linearly dependent if and only if
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The determinant of a matrix of order
over a field is equal to 1 if and only if
is the product of elementary matrices of the form
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where , while
is a matrix with its only non-zero entries equal to 1 and positioned at
.
The theory of determinants was developed in relation to the problem of solving systems of linear equations:
![]() | (2) |
where are elements of the field
. If
, where
is the matrix of the system (2), then this system has a unique solution, which can be calculated by Cramer's formulas (see Cramer rule). When the system (2) is given over a ring
and
is invertible in
, the system also has a unique solution, also given by Cramer's formulas.
A theory of determinants has also been constructed for matrices over non-commutative associative skew-fields. The determinant of a matrix over a skew-field (the Dieudonné determinant) is introduced in the following way. The skew-field
is considered as a semi-group, and its commutative homomorphic image
is formed.
is a group,
, with added zero 0, while the role of
is taken by the group
with added zero
, where
is the quotient group of
by the commutator subgroup. The epimorphism
,
, is given by the canonical epimorphism of groups
and by the condition
. Clearly,
is the unit of the semi-group
.
The theory of determinants over a skew-field is based on the following theorem: There exists a unique mapping
![]() |
satisfying the following three axioms:
I) if the matrix is obtained from the matrix
by multiplying one row from the left by
, then
;
II) if is obtained from
by replacing a row
by a row
, where
, then
;
III) .
The element is called the determinant of
and is written as
. For a commutative skew-field, axioms I), II) and III) coincide with conditions
), 2) and 3), respectively, and, consequently, in this instance ordinary determinants over a field are obtained. If
, then
; thus, the mapping
is surjective. A matrix
from
is invertible if and only if
. The equation
holds. As in the commutative case,
will not change if a row
of
is replaced by a row
, where
,
. If
,
if and only if
is the product of elementary matrices of the form
,
,
. If
, then
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Unlike the commutative case, does not have to coincide with
. For example, for the matrix
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over the skew-field of quaternions (cf. Quaternion), , while
.
Infinite determinants, i.e. determinants of infinite matrices, are defined as the limit towards which the determinant of a finite submatrix converges when its order is growing infinitely. If this limit exists, the determinant is called convergent; in the opposite case it is called divergent.
The concept of a determinant goes back to G. Leibniz (1678). H. Cramer was the first to publish on the subject (1750). The theory of determinants is based on the work of A. Vandermonde, P. Laplace, A.L. Cauchy and C.G.J. Jacobi. The term "determinant" was first coined by C.F. Gauss (1801). The modern meaning was introduced by A. Cayley (1841).
References
[1] | A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) |
[2] | A.I. Kostrikin, "Introduction to algebra" , Springer (1982) (Translated from Russian) |
[3] | N.V. Efimov, E.R. Rozendorn, "Linear algebra and multi-dimensional geometry" , Moscow (1970) (In Russian) |
[4] | R.I. Tyshkevich, A.S. Fedenko, "Linear algebra and analytic geometry" , Minsk (1976) (In Russian) |
[5] | E. Artin, "Geometric algebra" , Interscience (1957) |
[6] | N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) |
[7] | V.F. Kagan, "Foundations of the theory of determinants" , Odessa (1922) (In Russian) |
Comments
References
[a1] | J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) |
[a2] | K. Hoffman, R. Kunze, "Linear algebra" , Prentice-Hall (1961) |
[a3] | M. Koecher, "Lineare Algebra und analytische Geometrie" , Springer (1983) |
[a4] | S. Lang, "Linear algebra" , Addison-Wesley (1970) |
Determinant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Determinant&oldid=12692