Let X, Y be two sequence spaces defined by speeds of the convergence, i.e.; by monotonically increasing positive sequences. In this paper, we give necessary and sufficient conditions for a matrix A (with real or complex entries) to map X into Y. Also the analogue of the well known result of Steinhaus, which states that a regular matrix cannot transform each bounded sequence into convergent sequence, for the sequence spaces defined by the speeds of convergence has been proved.