  In this paper, we study matricial representations of certain finitely
presented groups with N-generators of order-2. As an application, we consider a
group algebra under our representations. Specifically, we characterize the
inverses of all group elements in terms of matrices in the group algebra. From
the study of this characterization, we realize there are close relations
between the trace of the radial operator and the Lucas numbers appearing in the
Lucas triangle.
