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Chapter 4: Q10E (page 191)

Question 10: Determine if \(P = \left[ {\begin{array}{*{20}{c}}1&{.2}\\0&{.8}\end{array}} \right]\) is a regular stochastic matrix.

Short Answer

Expert verified

\(P\)is not a regular stochastic matrix.

Step by step solution

01

Define a regular stochastic matrix

The stochastic matrix isregular if some matrix power \({p^k}\) contains only positive entries.

02

 Step 2: Determine whether the matrix is regular stochastic

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

\(\begin{aligned}{}{P^2} &= \left( {\begin{aligned}{{}}1&{.2}\\0&{.8}\end{aligned}} \right)\left( {\begin{aligned}{{}}1&{.2}\\0&{.8}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}{1 + 0}&{0.2 + 0.16}\\{0 + 0}&{0 + 0.64}\end{aligned}} \right)\\ &= \left( {\begin{aligned}{{}}1&{.36}\\0&{.64}\end{aligned}} \right)\end{aligned}\)

\({P^k} = \left( {\begin{aligned}{{}}1&{1 - {8^k}}\\0&{{{.8}^k}}\end{aligned}} \right)\)haszero as its \(\left( {2,1} \right)\) entry for all \(k\).

Thus, \(P\) is not a regular stochastic matrix.

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