In the introduction we mentioned that one can formulate much of the ordinary differential geometry of a manifold in terms of the algebra of smooth functions defined on it. It is possible to define finite noncommutative geometries by replacing this algebra with the algebra Mn of n × n complex matrices. Since Mn is of finite dimension all calculations reduce to pure algebra. The Hodge-de Rham theorem, for example, becomes an easy result on the range and the kernel of a symmetric matrix. Matrix geometry is also interesting in being similar to the ordinary geometry of compact Lie groups. It constitutes therefore a transition to the more abstract formalism of general noncommutative geometry which is given in the next chapter. We shall show in Chapter 7 that there is a sense in which the ordinary geometry of the torus and the sphere can be considered as the limit of a sequence of matrix geometries. In this chapter we shall be exclusively interested in matrix algebras. When however a result has nothing to do with the dimension of the algebra and is valid in more general contexts we shall often designate the algebra by A in order to avoid an unnecessary repetition of formulae in the next chapter. It is worth emphasizing also that, as we shall see in Section 4.3, many interesting algebras with involutions can be obtained as limits of matrix algebras. The present chapter can be read and understood as algebra without any knowledge of geometry but a certain familiarity with this subject is necessary as motivation. We gave in Chapter 2 a brief introduction.