Question: A small town has two public library branches, A and B. Patrons checking out books can return them to any of the branches, where the

A small town has two public library branches, A and B. Patrons checking out books can return them to any of the branches, where the books stay until checked out again. Suppose that of the books borrowed from A, 40% are returned to B. Of the books borrowed from B, 50% are returned to A.

(A) Find the stochastic matrix for this situation. (solution)

(B) If a book is borrowed from A, what is the probability that it ends up at B after 5 more circulations? Use the idea of eigenvalues and eigenvectors. If you just multiply a matrix 5 times using a calculator in order to get your answer, you will get no credit for this problem at all.

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