stylistic:tone
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 R {\displaystyle R}, such as the integers Z {\displaystyle \mathbf {Z} }, as opposed to the field of real or complex numbers. Moreover, the characterization of the determinant as the unique alternating multilinear map that satisfies det (I) = 1 {\displaystyle \det(I)=1} still holds, as do all the properties that result from that characterization.

A matrix A ∈ Mat n × n (R) {\displaystyle A\in \operatorname {Mat} _{n\times n}(R)} is invertible (in the sense that there is an inverse matrix whose entries are in R {\displaystyle R}) if and only if its determinant is an invertible element in R {\displaystyle R}. For R = Z {\displaystyle R=\mathbf {Z} }, this means that the determinant is +1 or −1. Such a matrix is called unimodular.

The determinant being multiplicative, it defines a group homomorphism

between the general linear group (the group of invertible n × n {\displaystyle n\times n} -matrices with entries in R {\displaystyle R}) and the multiplicative group of units in R {\displaystyle R}. Since it respects the multiplication in both groups, this map is a group homomorphism.

Given a ring homomorphism f: R → S {\displaystyle f:R\to S}, there is a map GL n (f): GL n (R) → GL n (S) {\displaystyle \operatorname {GL} _{n}(f):\operatorname {GL} _{n}(R)\to \operatorname {GL} _{n}(S)} given by replacing all entries in R {\displaystyle R} by their images under f {\displaystyle f}. The determinant respects these maps, i.e., the identity

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 m {\displaystyle m} of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo m {\displaystyle m} (the latter determinant being computed using modular arithmetic). In the language of category theory, the determinant is a natural transformation between the two functors GL n {\displaystyle \operatorname {GL} _{n}} and (−) × {\displaystyle (-)^{\times }}. Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of algebraic groups, from the general linear group to the multiplicative group,