How can we predict where a system will end up in the long run if it moves between states with fixed probabilities? Will a population grow or shrink, and what will its age groups look like? If different parts of an economy all need each other’s output, how much must each produce to satisfy final demand? All these questions involve nonnegative matrices — matrices whose entries are never negative. Often these matrices have a special largest eigenvalue that holds the answer: the steady state, the growth rate, or the economy’s productive capacity. In this chapter we learn how to find that number and why it works so reliably.