A market analysis of car purchasing trends in a certain region has concluded that a family purchases a new car once every three years. They found that a small car is replaced with another small car 80% of the time and that a large car is replaced with another large car 60% of the time.

(a) Define a transition matrix P for a Markov chain that describes the buying patterns. (Note that each “step” here is a 3-year period.)

(b) Find the N-step transition matrix corresponding to 12 years from now.

(c) If there are currently 40,000 small cars and 50,000 large cars in the region today, what is the prediction for the number of each in 12 years’ time?

(d) Does the distribution of small/large cars stabilize?
i. Find the eigenvalues of P.
ii. Diagonalizable P by finding an invertible matrix Q and diagonal matrix D such that P = QDQ−1.
iii. Use this to write P ^N and compute Lim N → ∞ P^N.

You’ll need the fact that Lim n→∞ (2/ 5)N = 0.

This gives stable proportions.