The linear algebraic theory of the Pascal and Vandermonde matrix is well developed by many authors. In the last two decades many interrelations have been discovered between the mentioned matrices, their generalizations and the Stirling matrices. We follow this direction and discover new matricial relations by using the so-called r-Whitney numbers. Along this way, we develop two natural extensions of the Vandermonde matrix with which we can study and evaluate successive power sums of arithmetic progressions and win new identities for the r-Whitney numbers.