This article introduces the (r, n)-min and (r, n)-max matrices, which generalize the concepts of the r-min, r-max, min, and max matrices. Subsequently, our focus is directed towards the discussion of the algebraic structure of the (r, n)-min and (r, n)-max matrices, their reciprocal matrices, and distinct representations. These efforts enable us to elucidate several mathematical properties, including determinants, inverses, and certain norms.