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Chapter 4: Q9E (page 191)

Question 9: Determine if \(P = \left[ {\begin{array}{*{20}{c}}{.2}&1\\{.8}&0\end{array}} \right]\) is a regular stochastic matrix.

Short Answer

Expert verified

\(P\) is a regular stochastic matrix.

Step by step solution

01

Regular stochastic matrix

The stochastic matrix is regularif some matrix power \({p^k}\) contains only strictly positive entries.

02

Determine whether the matrix is regular stochastic

Compute the matrix \({P^2}\) as shown below:

\(\begin{aligned}{}{P^2} &= \left( {\begin{aligned}{{}}{.2}&1\\{.8}&0\end{aligned}} \right)\left( {\begin{aligned}{{}}{.2}&1\\{.8}&0\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}{0.04 + 0.8}&{0.2 + 0}\\{0.16 + 0}&{0.8 + 0}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}{0.84}&{0.2}\\{0.16}&{0.8}\end{aligned}} \right)\end{aligned}\)

Every entry in \({P^2}\) is strictly positive.

Thus, \(P\) is a regular stochastic matrix.

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