Let $A,B$ and $M$ be matrices with real or complex entries. 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\`aro matrix and $A$ is a normal matrix, is described.