 One of the important uses for matrices is describing what's called a Leslie model. In 1945, the Scottish mathematician Patrick Holt Leslie pointed out that we can use transition matrices to describe populations where individuals can be classified according to life stages. For example, butterflies go through four stages, egg, larva, pupa, and adult. Humans go through several stages, infant, toddler, child, adolescent, adult. Products go through several life stages, design, beta, release, recall, and obsolescence. And any time we can describe how an individual moves through a set of stages, we can use a Leslie model to describe the population. So for example, suppose I have an insect population that grows according to the following rules, 10% of the eggs hatched into larvae, the rest are presumed eaten. 20% of the larva mature into pupa, the rest are presumed eaten. 10% of the pupa mature into adults. You get the idea. 50% of the adults survive one time cycle. And then every adult produces 100 eggs. So let's construct a transition matrix for this population of insects. In order to do that, we'll want to set up a vector that represents our population. And since we have our four life stages, the components of our vector will be the number of individuals in those four life stages. And so my vector gives me the number of eggs, larva, pupae, and adult at some time n. And so our task is, given that we know the number of individuals in each of these four life stages at a particular point in time, that is the number of individuals in each of those life stages at the next point in time. In other words, I want formulas for the number of eggs, larva, pupae, and adult at the next point in time. Now a useful rule in life and in mathematics is that it's easier to figure out where you've come from than it is to figure out where you're going. So let's take a look at the number of eggs. If we look at how our insects live, we see that the only way that we get eggs is from the adults. And every adult produces 100 eggs, so the number of eggs that we will have is 100 times the number of adults that we currently have. Since our goal is to express the transition matrix and consistency counts, we should make sure that the number of individuals in each of these life stages is included in our formula, even if the coefficient has to be zero. Next, how about the number of larvae? And if we take a look at our description of how the insect population grows, we see that the only place we get larvae from are that 10% of the eggs hatch into larva. And so that means the number of larvae that we get is going to be 10.1 times the number of eggs. And again, we'll write this using all of our variables. How about the number of pupa? So the number of pupa are going to come from the larva, and 20% of the larva mature into pupa. And so that gives us the number of pupa, again, including all of our other terms with coefficient of zero as necessary. And finally, let's take a look at those adults. So the adults actually come from two different sources. 10% of the pupa mature into adults, but there's a second source. 50% of the existing adults survive. And so the number of adults at time n plus one is going to be 10% of the pupa and 50% of the adults. And again, we'll include our other terms using the zero coefficient. And our transition matrix is going to consist of the coefficients of these formulas.