In this paper, a study is made on two well-known operator matrices; the Rhaly operator R a and the generalized difference operator ∆ ab. Firstly, some compactness results for the operators R a and ∆ ab , whose purpose is to help in describing their spectra, are derived. Next, general results on investigating the spectra of such operators on a large class of Banach sequence spaces are established. These results give a complete description of the spectra. The obtained results unify, extend and improve many comparable results in the existing literature.