  The underlying algebra for a noncommutative geometry is taken to be a matrix
algebra, and the set of derivatives the adjoint of a subset of traceless
matrices. This is sufficient to calculate the dual 1-forms, and show that the
space of 1-forms is a free module over the algebra of matrices. The concept of
a generalised algebra is defined and it is shown that this is required in order
for the space of 2-forms to exist. The exterior derivative is generalised for
higher order forms and these are also shown to be free modules over the matrix
algebra. Examples of mappings that preserve the differential structure are
given. Also given are four examples of matrix generalised algebras, and the
corresponding noncommutative geometries, including the cases where the
generalised algebra corresponds to a representation of a Lie algebra or a
$q$-deformed algebra.
