 An important use of transition matrices occurs in what's called discrete time modeling. An example of this occurs in a rather famous problem posed by Leonardo of Pisa in the 13th century. So, suppose I start with a pair of rabbits, and then leave them alone for a while, and they mature into adult rabbits, but if I leave them alone for too long, they will produce another pair of rabbits. And so the question is, how many rabbits do we have after a certain period of time? In order to answer that question, we have to make some assumptions about how the rabbit population grows. So we'll make the following assumptions. First, we'll assume that rabbits mature after one month, and they produce a pair of rabbits every month thereafter. We'll also assume that the pairs are always a male and female rabbit, and that mature rabbits always find a mate, and then finally, we'll assume that no rabbits die. And so what we're interested in is we want to let xn and yn be the number of pairs of immature and mature rabbits, respectively, at the end of month n. And we want to construct the transition matrix that tells us how to go from the number of immature and mature pairs at the end of month n to the number of immature and mature pairs at the end of month n plus 1. Now it might not be obvious how to do that, so let's run our rabbits through a couple of months. So let's suppose at the start I have one pair of immature rabbits at the end of month 1. I turn my back, so to speak, and at the end of month 2, this pair of immature rabbits has become a pair of mature rabbits. Now I wait another month. And since I now have a pair of mature rabbits, they will produce a pair of immature rabbits. And so at the end of month 3, I have one pair of mature rabbits and one pair of immature rabbits. What if we go into month 4? First, the immature rabbits will become mature rabbits, and second, the mature rabbits will produce another pair of immature rabbits. And so now we have two pairs of mature rabbits and one pair of immature rabbits. If we go another month, the two mature pairs produce two more immature pairs. The immature pair becomes a mature pair. And so now we have three mature pairs and two immature pairs. And we can continue this logic as far as we want. But let's do a little bit of analysis. The transition matrix T will be the coefficients of the formulas that give us the values of xn plus 1 and yn plus 1 from the values of xn and yn. So a useful rule of life and in mathematics is that it's usually easier to figure out where you're coming from than where you're going to. So let's consider this. This xn plus 1 is going to be the number of pairs of immature rabbits at the end of month n plus 1. Now by our assumption, every pair of mature rabbits produces a pair of immature rabbits. So that means xn plus 1 will be equal to yn. And since we want to write this down as a transition matrix, that means our formula should be an equation in standard form, which means we'll include all of our variables xn and yn. And we'll use coefficients of 0 and 1 as necessary. So our equation becomes xn plus 1 is equal to 0xn plus 1 yn. And these coefficients will give us the first rule of our transition matrix. Similarly, yn plus 1 is the number of mature rabbits at the end of month n plus 1. And these rabbits come from two sources. First of all, the mature rabbits at the end of month n are still alive. We've assumed that no rabbits die. So that's yn rabbits. In addition, we've assumed that the immature rabbits will become mature after one month. So all of the immature rabbits at the end of month n are now mature rabbits. And so this adds xn to the total number of mature rabbits. And so yn plus 1 is going to be xn plus yn. And again, writing this formula in standard form, we have 1xn plus 1 yn equals yn plus 1. And these coefficients will give us the second row of the transition matrix.