This paper studies linear stochastic approximation(SA) algorithms and their application to multi-agent systems in engineering and sociology. As main contribution, we provide necessary and sufficient conditions for convergence of linear SA algorithms to a deterministic or random final vector. We also characterize the system convergence rate, when the system is convergent. Moreover, differing from non-negative gain functions in traditional SA algorithms, this paper considers also the case when the gain functions are allowed to take arbitrary real numbers. Using our general treatment, we provide necessary and sufficient conditions to reach consensus and group consensus for first-order discrete-time multi-agent system over random signed networks and with state-dependent noise. Finally, we extend our results to the setting of multi-dimensional linear SA algorithms and characterize the behavior of the multi-dimensional Friedkin-Johnsen model over random interaction networks.