Let A, B and M be matrices with real or complex entries. In this paper two classes of matrices M, transforming the summability domain of A into the summability domain of B, are characterized. The M-consistency of A and B on the summability domain of A for a triangular matrix A and a regular matrix B is considered. As an application of the main results of this paper one class of matrices M, transforming the summability domain of A into the summability domain of B in the special case if B is the Ces?ro matrix and A is a normal matrix, is described.