  The theorem of Hilbert- Burch provides a description of codimension two
determinantal varieties and their deformations in terms of their presentation
matrices. In this work we use this correspondence to study properties of
determinantal varieties, based on methods of singularity theory. We establish
the theory of singularities for nxp matrices extending previous results of
Bruce and Tari and Fr\"uhbis-Kr\"uger. The main result of this work is the
description of equivalent conditions to G-finite determinacy of the
presentation matrix of Cohen-Macaulay varieties of codimension 2. We apply the
results to obtain sufficient conditions for topological triviality of
deformations of weighted homogeneous matrices.
