Dynamical systems with abrupt changes or several structures or interruptions to communications between subsystems can be effectively modelled by random jump linear systems. Random jump linear systems are a kind of stochastic hybrid systems, which consists of a set of linear systems, also called multiple modes, and the switching among them is governed by a random jump process. In particular, we consider that the underlying random jump process is time inhomogeneous. Roughly, a time inhomogeneous random jump process is one that has time varying transition probabilities (or rates) among its modes. It is a practical consideration, because, in general, the physical systems are time inhomogeneous across its different properties and transition probabilities (or rates) among its modes can also be viewed as a time inhomogeneous phenomenon.