Chapter 4.9, Problem 19E

Let S be the 1 × n row matrix with a 1 in each column, $s=\left[\begin{array}{cccc}1& 1& \cdots & 1\end{array}\right]$

1. a. Explain why a vector x in ℝⁿ is a probability vector if and only if its entries are nonnegative and Sx = 1. (A 1 × 1 matrix such as the product Sx is usually written without the matrix bracket symbols.)
2. b. Let P be an n × n stochastic matrix. Explain why SP = S.
3. c. Let P be an n × n stochastic matrix, and let x be a probability vector. Show that Px is also a probability vector.