These are matrices whether the individual elements are in the form of probabilities. The probabilities signify the probability of one event following another event that is the probability of transition from one event to the next.
The probabilities of the different changes applied to the initial state by matrix multiplication, provide a forecast of the succeeding state. Usually a transition matrix is defined along with its columns adding upto one and state vectors as column vectors. In this type of case the succeeding state is found by pre-multiplying the transition matrix by the proceeding state as column vector
If the transition matrices are given along with their rows adding up to one so then the succeeding state is found by post multiplying the transition matrix, by the preceding state as row vector.