**Title: Understanding Steady-State Vectors in Transition Matrices**---**Find the steady-state vector associated with the given transition matrix. (Give exact answers. Do not round.)**\[\begin{bmatrix}0.2 & 0.8 \\0.1 & 0.9 \\\end{bmatrix}\]This matrix represents a transition matrix often used in Markov processes. Each element in the matrix represents the probability of transitioning from one state to another. For example, in this matrix:- 0.2 is the probability of staying in the first state.- 0.8 is the probability of transitioning from the first state to the second state.- 0.1 is the probability of transitioning from the second state to the first state.- 0.9 is the probability of staying in the second state.The goal is to find the steady-state vector, a vector that remains unchanged after applications of the transition matrix. This vector shows the long-term behavior of the system.Below is a layout representing an unanswered system of equations, depicted by blank brackets, to compute the steady-state vector:\[\begin{bmatrix}\text{\_\_\_} \\\text{\_\_\_} \\\end{bmatrix}\]Arrows indicate the direction of the equations, relating to inputs and transformations within the system.**Key Concepts:**1. **Transition Matrix**: A square matrix used to describe the transitions of a Markov chain. Each of its rows is a probability vector, meaning all elements are non-negative and each row sums to one. 2. **Steady-State Vector**: A probability vector, denoted usually by $\pi$, that becomes stable, satisfying the condition $\pi = \pi P$, where $P$ is the transition matrix.Understanding these concepts allows students to explore how systems evolve over time and reach equilibrium, which is integral in fields such as economics, genetics, and computer science.---

Given,T=0.20.80.10.9Let L =xyTL=L0.20.80.10.9xy=xyx+y=10.2x+0.8y=x0.1x+0.9y=yRe-write the…

Answered: Find the steady-state vector associated…

Find the steady-state vector associated with the given transition matrix. (Give exact answers. Do not round.) 0.2 0.8 0.1 0.9

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Title: Understanding Steady-State Vectors in Transition Matrices**

---

**Find the steady-state vector associated with the given transition matrix. (Give exact answers. Do not round.)**

\[
\begin{bmatrix}
0.2 & 0.8 \\
0.1 & 0.9 \\
\end{bmatrix}
\]

This matrix represents a transition matrix often used in Markov processes. Each element in the matrix represents the probability of transitioning from one state to another. For example, in this matrix:

- 0.2 is the probability of staying in the first state.
- 0.8 is the probability of transitioning from the first state to the second state.
- 0.1 is the probability of transitioning from the second state to the first state.
- 0.9 is the probability of staying in the second state.

The goal is to find the steady-state vector, a vector that remains unchanged after applications of the transition matrix. This vector shows the long-term behavior of the system.

Below is a layout representing an unanswered system of equations, depicted by blank brackets, to compute the steady-state vector:

\[
\begin{bmatrix}
\text{\_\_\_} \\
\text{\_\_\_} \\
\end{bmatrix}
\]

Arrows indicate the direction of the equations, relating to inputs and transformations within the system.

**Key Concepts:**

1. **Transition Matrix**: A square matrix used to describe the transitions of a Markov chain. Each of its rows is a probability vector, meaning all elements are non-negative and each row sums to one.

2. **Steady-State Vector**: A probability vector, denoted usually by $\pi$, that becomes stable, satisfying the condition $\pi = \pi P$, where $P$ is the transition matrix.

Understanding these concepts allows students to explore how systems evolve over time and reach equilibrium, which is integral in fields such as economics, genetics, and computer science.

---

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