The set of square matrices of dimension n form an algebraic entity known as a ring. By definition, a ring consists of a set R and two operators (addition + and multiplication ×) such that
R is an Abelian group with respect to addition.
R is a semigroup with respect to multiplication.
R is left distributive, i.e. a × (b + c) = (a × b) + (a × c).
R is right distributive, i.e.(b + c) × a = (b × a) + (c × a).
An Abelian group consists of a set G and a binary operator such that
G is associative with respect to the operator.
G has an identity element with respect to the operator.
Each element of G has an inverse with respect to the operator.
G is commutative with respect to the operator.
A semigroup consists of a set G and a binary operator such that G is associative with respect to the operator.
For non-square matrices, even these limited properties are not generally true. The following sections examine the algebraic properties of matrices in further detail.