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Chapter 9: Problem 9

Determine which of the matrices are stochastic. \(\left[\begin{array}{rr}.2 & .3 \\ .3 & .1 \\ .5 & .6\end{array}\right]\)

Short Answer

Expert verified
The given matrix \(M=\begin{bmatrix} 0.2 & 0.3 \\ 0.3 & 0.1 \\ 0.5 & 0.6 \end{bmatrix}\) is not a square matrix since it has 3 rows and 2 columns. Thus, it cannot be a stochastic matrix.

Step by step solution

01

Identify given matrix M

First, let us identify the given matrix: \(M=\begin{bmatrix} 0.2 & 0.3 \\ 0.3 & 0.1 \\ 0.5 & 0.6 \end{bmatrix}\)
02

Check if the given matrix is square

To determine whether the given matrix is square or not, let us check if the number of rows is equal to the number of columns. Number of rows = 3 Number of columns = 2 Since the number of rows is not equal to the number of columns (3 鈮� 2), the matrix is not a square matrix.
03

Conclusion

Since the given matrix is not square, it cannot be a stochastic matrix. Therefore, the given matrix is not stochastic.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrices
Matrices are fundamental concepts in mathematics and computer science, essentially collections of numbers enclosed in a rectangular array. Typically, matrices help to represent and solve problems in linear algebra.
Matrices have dimensions that consist of rows and columns. For instance, a matrix with three rows and two columns is described as a 3x2 matrix.
A matrix is often written with square brackets or parentheses, and its elements are typically denoted by variables or numbers. Matrices can have varying purposes depending on their specific arrangement.
  • The elements of a matrix must be complex or real numbers.
  • Matrices are used for computations and transformations in graphics, statistics, and data analysis.
Understanding matrices is essential as it forms the basis of more advanced concepts such as transformations and operations in different fields of study.
Square Matrix
A square matrix is a particular type of matrix where the number of rows is equal to the number of columns. This equality gives the matrix a square shape, hence the name 'square matrix'.
For instance, matrices with dimensions like 2x2, 3x3, or 4x4 are square matrices.
Square matrices have several special properties and applications:
  • Determinants: You can compute a determinant for a square matrix, which is a scalar value useful in various mathematical calculations such as finding inverses or solving systems of linear equations.
  • Diagonal Matrix: A type of square matrix where all elements outside the main diagonal are zeroes.
  • Identity Matrix: A type of square matrix where the main diagonal elements are all ones, and the rest are zeros. This matrix acts like the number 1 in matrix multiplication.
Understanding how square matrices operate is vital in solving algebraic expressions and understanding transformations in vector spaces.
Matrix Types
Matrices come in various types, each with unique features and applications. It is crucial to identify these types to know how different matrices can be used.
Here are some common types of matrices:
  • Row Matrix: Consists of a single row.
  • Column Matrix: Consists of a single column.
  • Rectangular Matrix: A matrix where the number of rows is not equal to the number of columns, like 3x2 matrices.
  • Square Matrix: As explained, where rows equal columns, such as 2x2 or 3x3 matrices.
  • Diagonal Matrix: A square matrix with zeros everywhere except the main diagonal.
  • Stochastic Matrix: A square matrix where each column or row sums up to one, used for probability distributions.
Recognizing these matrix types helps with accurately applying operations and transformations in relevant mathematical or computational problems.

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Most popular questions from this chapter

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

Find the optimal strategies, \(P\) and \(Q\), for the row and column players, respectively. Also compute the expected payoff \(E\) of each matrix game and determine which player it favors, if any, if the row and column players use their optimal strategies. \(\left[\begin{array}{rr}2 & 5 \\ -2 & 4\end{array}\right]\)

Two dentists, Lydia Russell and Jerry Carlton, are planning to establish practices in a newly developed community. Both have allocated approximately the same total budget for advertising in the local newspaper and for the distribution of fliers announcing their practices. Because of the location of their offices, Russell will get \(48 \%\) of the business if both dentists advertise only in the local newspaper; if both dentists advertise through fliers, then Russell will get \(45 \%\) of the business; if Russell advertises exclusively in the local newspaper and Carlton advertises exclusively through fliers, then Russell will get \(65 \%\) of the business. Finally, if Russell advertises through fliers exclusively and Carlton advertises exclusively in the local newspaper, then Russell will get \(50 \%\) of the business. a. Construct the payoff matrix for the game and show that it is not strictly determined. b. Find the optimal strategy for both Russell and Carlton.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. A stochastic matrix \(T\) is a regular Markov chain if the powers of \(T\) approach a fixed matrix whose columns are all equal.

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