# 2 Matrix Algebra

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.