Some Markov chains have very useful characteristics. Specially in the case of credit risk, we are interested in concept such as the probability of default, the passing of time between an initial state and default and topics such as the setting where we have more than one Markov-driven economic agent and where the credit status of one may affect the others. We will see that for such special cases with a transition matrix that satisfies some well defined conditions, it is possible to compute the alluded probabilities with just a few matrix operations. We use examples to illustrate the application of the most commonly used theorems of Markov Chains.