“Selfing” is a plant breeding scheme that mates an organism with itself, selects one of the progeny randomly and mates it with itself, and so forth from generation to generation. Suppose at some genetic locus there are two alleles A and

a. A plant can have any of the three genotypes A/A, A/a, or a/a. Define a Markov chain with three states giving the genotype of the current plant in the selfing scheme. Show that the nth power of the transition matrix is P n = 

100 1 2 − ( 1 2 )n+1 ( 1 2 )n 1 2 − ( 1 2 )n+1 001 
 .
What is limn→∞ P n? Demonstrate that this Markov chain has multiple equilibrium distributions and characterize them.