Use the age transition matrix L and age distribution vector x₁ to find the age distribution vectors x₂ and X3. 0 36 1 00 X2 X3 || || L = x = t 0 000 ↓1 000 0 I X₁ = 18 18 18 Then find a stable age distribution vector. E
Use the age transition matrix L and age distribution vector x₁ to find the age distribution vectors x₂ and X3. 0 36 1 00 X2 X3 || || L = x = t 0 000 ↓1 000 0 I X₁ = 18 18 18 Then find a stable age distribution vector. E
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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![Use the age transition matrix \( L \) and age distribution vector \( \mathbf{x}_1 \) to find the age distribution vectors \( \mathbf{x}_2 \) and \( \mathbf{x}_3 \).
\[
L = \begin{bmatrix}
0 & 3 & 6 \\
1 & 0 & 0 \\
0 & \frac{1}{3} & 0
\end{bmatrix}, \quad \mathbf{x}_1 = \begin{bmatrix}
18 \\
18 \\
18
\end{bmatrix}
\]
\[
\mathbf{x}_2 = \begin{bmatrix}
\boxed{\phantom{0}} \\
\boxed{\phantom{0}} \\
\boxed{\phantom{0}}
\end{bmatrix}
\]
\[
\mathbf{x}_3 = \begin{bmatrix}
\boxed{\phantom{0}} \\
\boxed{\phantom{0}} \\
\boxed{\phantom{0}}
\end{bmatrix}
\]
Then find a stable age distribution vector.
\[
\mathbf{x} = t \begin{bmatrix}
\boxed{\phantom{0}} \\
\boxed{\phantom{0}} \\
\boxed{\phantom{0}}
\end{bmatrix}
\]
**Explanation of Diagram:**
The provided matrix \( L \) is a 3x3 transition matrix used to model changes in age distribution over time.
The vector \( \mathbf{x}_1 \) represents the initial age distribution with three equal age cohorts of size 18.
The blank vector boxes for \( \mathbf{x}_2 \), \( \mathbf{x}_3 \), and a stable age distribution vector \( \mathbf{x} \), denoted as \( t \times \) the vector, represent places to calculate subsequent age distributions and a stable distribution after sufficient time has passed, presumably by iterating the matrix multiplication.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61ff295b-afd4-43c3-8ed2-2cc17b9c2249%2F40994742-b1d8-462d-8927-aa29e8b6ee89%2F85l0u6j_processed.png&w=3840&q=75)
Transcribed Image Text:Use the age transition matrix \( L \) and age distribution vector \( \mathbf{x}_1 \) to find the age distribution vectors \( \mathbf{x}_2 \) and \( \mathbf{x}_3 \).
\[
L = \begin{bmatrix}
0 & 3 & 6 \\
1 & 0 & 0 \\
0 & \frac{1}{3} & 0
\end{bmatrix}, \quad \mathbf{x}_1 = \begin{bmatrix}
18 \\
18 \\
18
\end{bmatrix}
\]
\[
\mathbf{x}_2 = \begin{bmatrix}
\boxed{\phantom{0}} \\
\boxed{\phantom{0}} \\
\boxed{\phantom{0}}
\end{bmatrix}
\]
\[
\mathbf{x}_3 = \begin{bmatrix}
\boxed{\phantom{0}} \\
\boxed{\phantom{0}} \\
\boxed{\phantom{0}}
\end{bmatrix}
\]
Then find a stable age distribution vector.
\[
\mathbf{x} = t \begin{bmatrix}
\boxed{\phantom{0}} \\
\boxed{\phantom{0}} \\
\boxed{\phantom{0}}
\end{bmatrix}
\]
**Explanation of Diagram:**
The provided matrix \( L \) is a 3x3 transition matrix used to model changes in age distribution over time.
The vector \( \mathbf{x}_1 \) represents the initial age distribution with three equal age cohorts of size 18.
The blank vector boxes for \( \mathbf{x}_2 \), \( \mathbf{x}_3 \), and a stable age distribution vector \( \mathbf{x} \), denoted as \( t \times \) the vector, represent places to calculate subsequent age distributions and a stable distribution after sufficient time has passed, presumably by iterating the matrix multiplication.
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