Editing Matrix multiplication (section)

==Abstract algebra==
The definition of matrix product requires that the entries belong to a semiring, and does not require multiplication of elements of the semiring to be [[commutative property|commutative]]. In many applications, the matrix elements belong to a field, although the [[tropical semiring]] is also a common choice for graph [[shortest path]] problems.<ref>{{cite book|title=Randomized Algorithms|first1=Rajeev|last1=Motwani|author1-link=Rajeev Motwani|first2=Prabhakar|last2=Raghavan|author2-link=Prabhakar Raghavan|publisher=Cambridge University Press|year=1995|isbn=9780521474658|page=280|url=https://books.google.com/books?id=QKVY4mDivBEC&pg=PA280}}</ref> Even in the case of matrices over fields, the product is not commutative in general, although it is [[Associative property|associative]] and is [[Distributive property|distributive]] over [[matrix addition]]. The [[identity matrices]] (which are the [[square matrices]] whose entries are zero outside of the main diagonal and 1 on the main diagonal) are [[identity element]]s of the matrix product. It follows that the {{math|''n'' × ''n''}} matrices over a [[Ring (mathematics)|ring]] form a ring, which is noncommutative except if {{math|1=''n'' = 1}} and the ground ring is commutative.

A square matrix may have a [[multiplicative inverse]], called an [[inverse matrix]]. In the common case where the entries belong to a [[commutative ring]] {{mvar|R}}, a matrix has an inverse if and only if its [[determinant]] has a multiplicative inverse in {{mvar|R}}. The determinant of a product of square matrices is the product of the determinants of the factors. The {{math|''n'' × ''n''}} matrices that have an inverse form a [[group (mathematics)|group]] under matrix multiplication, the [[subgroup]]s of which are called [[matrix group]]s. Many classical groups (including all [[finite group]]s) are [[group isomorphism|isomorphic]] to matrix groups; this is the starting point of the theory of [[group representation]]s.

Matrices are the [[morphisms]] of a [[category (mathematics)|category]], the [[category of matrices]]. The objects are the [[natural number]]s that measure the size of matrices, and the composition of morphisms is matrix multiplication. The source of a morphism is the number of columns of the corresponding matrix, and the target is the number of rows.