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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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Most popular questions from this chapter

In Exercise 7, find the coordinate vector \({\left( x \right)_{\rm B}}\) of x relative to the given basis \({\rm B} = \left\{ {{b_{\bf{1}}},...,{b_n}} \right\}\).

7. \({b_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{1}}}\\{ - {\bf{3}}}\end{array}} \right),{b_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{3}}}\\{\bf{4}}\\{\bf{9}}\end{array}} \right),{b_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{ - {\bf{2}}}\\{\bf{4}}\end{array}} \right),x = \left( {\begin{array}{*{20}{c}}{\bf{8}}\\{ - {\bf{9}}}\\{\bf{6}}\end{array}} \right)\)

Suppose \(A\) is \(m \times n\)and \(b\) is in \({\mathbb{R}^m}\). What has to be true about the two numbers rank \(\left[ {A\,\,\,{\rm{b}}} \right]\) and \({\rm{rank}}\,A\) in order for the equation \(Ax = b\) to be consistent?

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation.

a. What is the dimension of range of T if T is one-to-one mapping? Explain.

b. What is the dimension of the kernel of T (see section 4.2) if T maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\)? Explain.

If the null space of an \({\bf{8}} \times {\bf{5}}\) matrix A is 2-dimensional, what is the dimension of the row space of A?

If a\({\bf{6}} \times {\bf{3}}\)matrix A has a rank 3, find dim Nul A, dim Row A, and rank\({A^T}\).
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