Most of the control theory is developed around time-invariant systems where the state matrix A consists of scalars which are not functions of time. However, many physical systems are naturally modeled with the elements of the state matrix A depending on time. One example is the dynamics of a missile. Time- varying systems also arise when non-linear systems are linearized about a trajectory. In this work, the state-transition matrix is studied for time-varying systems in order to reach a general solution. The computational effort is significantly more complicated that the time-invariant case. There are many different methods in the literature for finding the state-transition matrix and one of them is adopted. Finally, a case study of Missile Dynamics will be analyzed and simulated in MATLAB.