Editing Determinant (section)

=== Square matrices over commutative rings ===

The above definition of the determinant using the Leibniz rule holds works more generally when the entries of the matrix are elements of a [[commutative ring]] <math>R</math>, such as the integers <math>\mathbf Z</math>, as opposed to the [[field (mathematics)|field]] of real or complex numbers. Moreover, the characterization of the determinant as the unique alternating multilinear map that satisfies <math>\det(I) = 1</math> still holds, as do all the properties that result from that characterization.<ref>{{harvnb|Dummit|Foote|2004|loc=§11.4}}</ref>

A matrix <math>A \in \operatorname{Mat}_{n \times n}(R)</math> is invertible (in the sense that there is an inverse matrix whose entries are in <math>R</math>) if and only if its determinant is an [[Unit (ring theory)|invertible element]] in <math>R</math>.<ref>{{harvnb|Dummit|Foote|2004|loc=§11.4, Theorem 30}}</ref> For <math>R = \mathbf Z</math>, this means that the determinant is +1 or −1. Such a matrix is called [[unimodular matrix|unimodular]].

The determinant being multiplicative, it defines a [[group homomorphism]]
:<math>\operatorname{GL}_n(R) \rightarrow R^\times, </math>
between the [[general linear group]] (the group of invertible <math>n \times n</math>-matrices with entries in <math>R</math>) and the [[multiplicative group]] of units in <math>R</math>. Since it respects the multiplication in both groups, this map is a [[group homomorphism]].

[[Image:Determinant as a natural transformation.svg|300px|thumb|right|The determinant is a natural transformation.]]
Given a [[ring homomorphism]] <math>f : R \to S</math>, there is a map <math>\operatorname{GL}_n(f) : \operatorname{GL}_n(R) \to \operatorname{GL}_n(S)</math> given by replacing all entries in <math>R</math> by their images under <math>f</math>. The determinant respects these maps, i.e., the identity
:<math>f(\det((a_{i,j}))) = \det ((f(a_{i,j})))</math>
holds. In other words, the displayed commutative diagram commutes.

For example, the determinant of the [[complex conjugate]] of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo <math>m</math> of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo <math>m</math> (the latter determinant being computed using [[modular arithmetic]]). In the language of [[category theory]], the determinant is a [[natural transformation]] between the two functors <math>\operatorname{GL}_n</math> and <math>(-)^\times</math>.<ref>{{harvnb|Mac Lane|1998|loc=§I.4}}. See also ''{{section link|Natural transformation#Determinant}}''.</ref> Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of [[algebraic group]]s, from the general linear group to the [[multiplicative group]],
:<math>\det: \operatorname{GL}_n \to \mathbb G_m.</math>