Homework 5 - due Wednesday, July 25, in class

Each problem is worth 10 points.

The following problems are from the textbook:

section 6.1: #2, #22, #26
section 6.2: #15, #20, #26
section 6.3: #3, #7, #21

Additionally, solve the following problem:
A Markov matrix is one whose entries are non-negative real numbers such that the sum of all entries in each column are 1. (In practice, these matrices arise when one is considering probabilities of a system moving from one of n states to another.) For a 2x2 Markov matrix, show that 1 is an eigenvalue, and that the other eigenvalue lies between -1 and 1 inclusive. This is a special case of a more general result which says that a Markov matrix always has 1 as an eigenvalue, and the other eigenvalues are at most 1 in modulus.