In this paper, we characterize the classes $((\ell_1)_{T'}(\ell_1)_{\widetilde{T}})$ and $(c_T,c_{\widetilde{T}})$ where $T=(t_{nk})^\infty_{n,k=0}$ and $\widetilde{T}=(\tilde{t}_{nk})^\infty_{n,k=0}$ are arbitrary triangles. We establish identities or estimates for the Hausdor measure of noncompactness of operators given by matrices in the classes $((\ell_1)_T,(\ell_1)\widetilde{T})$ and $(c_T,\widetilde{c}_T)$. Furthermore we give sufficient conditions for such matrix operators to be Fredholm operators on $(\ell_1)_T$ and $c_T$. As an application of our results, we consider the class $(bv,bv)$ and the corresponding classes of matrix operators. Our results are complementary to those in [2] and some of them are generalization for those in [3].