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
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:
is subject to the following three conditions:
1) is a linear function of any row of :
2) if the matrix is obtained from by replacing a row by a row , , then ;
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
where and are square matrices,
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;
when , ; for and from ,
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:
Let , and let be the cofactor of the entry . The following formulas are then true:
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
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
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
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:
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 ;
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
Unlike the commutative case, does not have to coincide with . For example, for the matrix
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).
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Determinant. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Determinant&oldid=12692