Q24E For the matrices A in Exercises ... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q24E (page 355)

For the matrices A in Exercises 20 through 24 find $\lim_{t 鈫� 鈭�} A^{t}$ . Feel free to use Theorem 7.4.1
$A = [\begin{matrix} 0.3 & 0.1 & 0.3 \\ 0.4 & 0.6 & 0.4 \\ 0.3 & 0.3 & 0.3 \end{matrix}]$

Short Answer

Expert verified

The value of $\lim_{t 鈫� 鈭�} A^{t} = [\begin{matrix} \frac{1}{5} & \frac{1}{5} & \frac{1}{5} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{3}{10} & \frac{3}{10} & \frac{3}{10} \end{matrix}]$

Step by step solution

Substituting the given matrix in Ax=x

$A x = x [\begin{matrix} 0.3 & 0.1 & 0.3 \\ 0.4 & 0.6 & 0.4 \\ 0.3 & 0.3 & 0.3 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}]$

Finding the equilibrium distribution xequ of A

Equating the values in the matrices and finding the values.

$0.3 x_{1} + 0.1 x_{2} + 0.3 x_{3} = x_{1}, 0.4 x_{1} + 0.6 x_{2} + 0.4 x_{3} = x_{2}, x_{1} + x_{2} + x_{3} = 1 x_{1} = \frac{1}{5}, x_{2} = \frac{1}{2}, x_{3} = \frac{3}{10} x_{e q u} = [\begin{matrix} \frac{1}{5} \\ \frac{1}{2} \\ \frac{3}{10} \end{matrix}]$

The value of $\lim_{t 鈫� 鈭�} A^{t}$ is,

$\lim_{t 鈫� 鈭�} A^{t} = [\begin{matrix} \frac{1}{5} & \frac{1}{5} & \frac{1}{5} \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{3}{10} & \frac{3}{10} & \frac{3}{10} \end{matrix}]$

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91影视

For the matrices A in Exercises 20 through 24 find $\lim_{t 鈫� 鈭�} A^{t}$ . Feel free to use Theorem 7.4.1
$A = [\begin{matrix} 0.3 & 0.1 & 0.3 \\ 0.4 & 0.6 & 0.4 \\ 0.3 & 0.3 & 0.3 \end{matrix}]$

Short Answer

Step by step solution

Substituting the given matrix in Ax=x

Finding the equilibrium distribution xequ of A

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91影视

For the matrices A in Exercises 20 through 24 findlimt鈫�鈭�At. Feel free to use Theorem 7.4.1A=[0.30.10.30.40.60.40.30.30.3]

Short Answer

Step by step solution

Substituting the given matrix in Ax=x

Finding the equilibrium distribution xequ of A

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Decision Maths

Mechanics Maths

Discrete Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

For the matrices A in Exercises 20 through 24 find $\lim_{t 鈫� 鈭�} A^{t}$ . Feel free to use Theorem 7.4.1
$A = [\begin{matrix} 0.3 & 0.1 & 0.3 \\ 0.4 & 0.6 & 0.4 \\ 0.3 & 0.3 & 0.3 \end{matrix}]$