<P> More generally, the set of m × n matrices can be used to represent the R - linear maps between the free modules R and R for an arbitrary ring R with unity . When n = m composition of these maps is possible, and this gives rise to the matrix ring of n × n matrices representing the endomorphism ring of R . </P> <P> A group is a mathematical structure consisting of a set of objects together with a binary operation, that is, an operation combining any two objects to a third, subject to certain requirements . A group in which the objects are matrices and the group operation is matrix multiplication is called a matrix group . Since in a group every element has to be invertible, the most general matrix groups are the groups of all invertible matrices of a given size, called the general linear groups . </P> <P> Any property of matrices that is preserved under matrix products and inverses can be used to define further matrix groups . For example, matrices with a given size and with a determinant of 1 form a subgroup of (that is, a smaller group contained in) their general linear group, called a special linear group . Orthogonal matrices, determined by the condition </P> <Dl> <Dd> M M = I, </Dd> </Dl>

When were matrices first used to describe transformations