We study the formation of the conductive channels in α-Si memristors and demonstrate their operation in the crossbar array. The latter can be utilised as the basic component of the neuromorphic chip tailored for edge computing. The conductive channels in α-Si are formed by the migration of Ag along with Cu ions. Such a channel has switching current-voltage characteristics at high bias, Vbias > 2V, and highly non-linear that at low bias, Vbias < 0.5V. Memristor can be re-programmed to different resistance states with short voltage pulses of amplitude above 2 V. We demonstrate the programming of the memristor crossbar array and its operation in vector-by-matrix multiplication with an 87% accuracy.