Consider an experiment with n possible states. At each step, individuals transition from state i to state j with probability sij. (This sort of model is useful in computer science, economics, statistics, and other areas.) Let S be the matrix with entries sij. This matrix satisfies: (i) all entries are ≥ 0, and (ii) all rows sum to 1. Matrices satisfying properties (i) and (ii) are called stochastic matrices. An example is ( (a) For the given example S, find S² and S3, and verify that they are also stochastic S= 2

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a.

From the given information,

$S=\left(\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ 0& 1\end{array}\right)$

Consider,

$$\begin{gathered} S^2=S*S \\ =\left(\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ 0& 1\end{array}\right)*\left(\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ 0& 1\end{array}\right) \\ =\left(\begin{array}{cc}\frac{1}{4}& \frac{1}{4}+\frac{1}{2}\\ 0& 1\end{array}\right) \\ =\left(\begin{array}{cc}\frac{1}{4}& \frac{3}{4}\\ 0& 1\end{array}\right) \end{gathered}$$

Here, all the entries are >=0 and the sum of the row elements is equal to 1.

That is, the conditions for the stochastic matrix are satisfied.

Hence, the matrix S^2 is also stochastic matrix.

$$\begin{gathered} S^3=S^2*S \\ =\left(\begin{array}{cc}\frac{1}{4}& \frac{3}{4}\\ 0& 1\end{array}\right)*\left(\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ 0& 1\end{array}\right) \\ =\left(\begin{array}{cc}\frac{1}{8}& \frac{1}{8}+\frac{3}{4}\\ 0& 1\end{array}\right) \\ =\left(\begin{array}{cc}\frac{1}{8}& \frac{7}{8}\\ 0& 1\end{array}\right) \end{gathered}$$

Here, all the entries are >=0 and the sum of the row elements is equal to 1.

That is, the conditions for the stochastic matrix are satisfied.

Hence, the matrix S^3 is also stochastic matrix.