Open in App
Log In Start studying!

Select your language

Suggested languages for you:

Problem 1

Determine whether the matrix is an absorbing stochastic matrix. \(\left[\begin{array}{ll}\frac{2}{5} & 0 \\ \frac{3}{5} & 1\end{array}\right]\)

Short Answer

Expert verified
The given matrix \(\left[\begin{array}{ll}\frac{2}{5} & 0 \\\ \frac{3}{5} & 1\end{array}\right]\) is a square matrix that has non-negative elements. However, the sum of each row is not equal to 1 (\(\frac{2}{5}\) for row 1 and \(\frac{8}{5}\) for row 2). As it does not fulfill the conditions to be a stochastic matrix, it cannot be an absorbing stochastic matrix.
See the step by step solution

Step by step solution

Unlock all solutions

Get unlimited access to millions of textbook solutions with Vaia Premium

Over 22 million students worldwide already upgrade their learning with Vaia!

Step 1: Check if it is a square matrix

The given matrix has 2 rows and 2 columns, so it is a square matrix.

Step 2: Check if all elements are non-negative

In the given matrix, all elements are non-negative (fractions or whole numbers).

Step 3: Check if the sum of each row is equal to 1

Compute the sum of each row: - Row 1: \(\frac{2}{5} + 0 = \frac{2}{5}\) - Row 2: \(\frac{3}{5} + 1 = \frac{8}{5}\) The sum of Row 1 is not equal to 1, and the sum of Row 2 is also not equal to 1. So the given matrix is not a stochastic matrix. Since it is not a stochastic matrix, there is no need to check further for condition 4. The given matrix does not meet all the conditions; therefore, it is not an absorbing stochastic matrix.

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Access millions of textbook solutions in one place

  • Access over 3 million high quality textbook solutions
  • Access our popular flashcard, quiz, mock-exam and notes features
  • Access our smart AI features to upgrade your learning
Get Vaia Premium now
Access millions of textbook solutions in one place

Most popular questions from this chapter

Chapter 9

Determine whether the matrix is an absorbing stochastic matrix. $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & .7 & .2 \\ 0 & .3 & .8\end{array}\right]$

Chapter 9

Within a large metropolitan area, \(20 \%\) of the commuters currently use the public transportation system, whereas the remaining \(80 \%\) commute via automobile. The city has recently revitalized and expanded its public transportation system. It is expected that 6 mo from now \(30 \%\) of those who are now commuting to work via automobile will switch to public transportation, and \(70 \%\) will continue to commute via automobile. At the same time, it is expected that \(20 \%\) of those now using public transportation will commute via automobile and \(80 \%\) will continue to use public transportation. a. Construct the transition matrix for the Markov chain that describes the change in the mode of transportation used by these commuters. b. Find the initial distribution vector for this Markov chain. c. What percentage of the commuters are expected to use public transportation \(6 \mathrm{mo}\) from now?

Chapter 9

Brady's, a conventional department store, and ValueMart, a discount department store, are each considering opening new stores at one of two possible sites: the Civic Center and North Shore Plaza. The strategies available to the management of each store are given in the following payoff matrix, where each entry represents the amounts (in hundreds of thousands of dollars) either gained or lost by one business from or to the other as a result of the sites selected. a. Show that the game is strictly determined. b. What is the value of the game? c. Determine the best strategy for the management of each store (that is, determine the ideal locations for each store).

Chapter 9

Rewrite each absorbing stochastic matrix so that the absorbing states appear first, partition the resulting matrix, and identify the submatrices \(R\) and \(S\). $\left[\begin{array}{lll}0 & .2 & 0 \\ .5 & .4 & 0 \\ .5 & .4 & 1\end{array}\right]$

Chapter 9

Over the years, consumers are turning more and more to newer and much improved videorecording devices. The following transition matrix describes the Markov chain associated with this process. Here \(\mathrm{V}\) stands for VHS recorders, D stands for DVD recorders, and \(\mathrm{H}\) stands for high definition video recorders. $A=\begin{array}{l}\mathrm{V} \\ \mathrm{D} \\\ \mathrm{H}\end{array}\left[\begin{array}{rrr}.10 & 0 & 0 \\ .70 & .60 & 0 \\\ .20 & .40 & 1\end{array}\right]$ a. Show that \(A\) is an absorbing stochastic matrix and rewrite it so that the absorbing state appears first. Partition the resulting matrix and identify the submatrices \(R\) and \(S\). b. Compute the steady-state matrix of \(A\) and interpret your results.

Join over 22 million students in learning with our Vaia App

The first learning app that truly has everything you need to ace your exams in one place.

  • Flashcards & Quizzes
  • AI Study Assistant
  • Smart Note-Taking
  • Mock-Exams
  • Study Planner
Join over 22 million students in learning with our Vaia App Join over 22 million students in learning with our Vaia App

Recommended explanations on Math Textbooks