The main purpose of this paper is to determine the fine spectrum with respect to Goldberg's classification of the operator defined by the lambda matrix over the sequence spaces 0 and c. As a new development, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator Λ on the sequence spaces 0 and c. Finally, we present a Mercerian theorem. Since the matrix Λ is reduced to a regular matrix depending on the choice of the sequence ( ) having certain properties and its spectrum is firstly investigated, our work is new and the results are comprehensive.