In this paper we deal with some new properties of the operator of first difference represented by the infinite matrix $\Delta$. We study the operator represented by the perturbed matrix $\Delta'_{pq}(a')$ obtained from $\Delta$ by changing one element. Then we give necessary and sufficient conditions for a matrix $A$ to map $s_\alpha((\Delta'_{pq}(a'))^\mu)$ into $s_\beta$, $\mu$ being an integer.