This article considers a class of finite-dimensional linear impulsive time-varying systems for which various sufficient and necessary algebraic criteria for complete controllability, including matrix rank conditions are established. The obtained controllability results are further synthesised for the time-invariant case, and under some special conditions on the system parameters, we obtain a Popov-Belevitch-Hautus (PBH)-type rank condition which employs eigenvalues of the system matrix for the investigation of their controllability. Numerical examples are provided that demonstrate--for the linear impulsive systems, null controllability need not imply their complete controllability, unlike for the non-impulsive linear systems.