An auto insurance company classifies its customers in three categories: poor, satisfactory, and preferred. No one moves from poor to preferred or from preferred to poor in 1 year. 40% of the customers in the poor category become satisfactory, 30% of those in the satisfactory category move to preferred, while 10% become poor; 20% of those in the preferred category are downgraded to satisfactory.

(a) Write the transition matrix for the model.

(b) What is the limiting fraction of drivers in each of these categories?