problem and find its transpose. Use "f" for the transpose. Minimize g = 4y₁ + 3y2 subject to 31 + y2 > 17 Y1 + 2y2 ≥ 22 31 > 0, 32 > 0
problem and find its transpose. Use "f" for the transpose. Minimize g = 4y₁ + 3y2 subject to 31 + y2 > 17 Y1 + 2y2 ≥ 22 31 > 0, 32 > 0
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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![### Forming the Matrix for a Minimization Problem
In this exercise, we aim to construct the matrix associated with a given minimization problem and find its transpose. The transpose will be represented using "f".
#### Problem Statement
Minimize the function \( g = 4y_1 + 3y_2 \) subject to the constraints:
\[
\begin{align*}
y_1 + y_2 & \geq 17 \\
y_1 + 2y_2 & \geq 22 \\
y_1 & \geq 0, \\
y_2 & \geq 0
\end{align*}
\]
#### Constructing the Matrix
The constraints can be expressed in matrix form:
- The matrix form involves arranging the coefficients of the variables from the inequalities and objective function.
- The matrix to be filled should include the coefficients and right-side constants from the constraints.
**Matrix Representation:**
\[
\begin{bmatrix}
1 & 1 \\
1 & 2
\end{bmatrix}
\quad
\begin{bmatrix}
17 \\
22
\end{bmatrix}
\]
#### Finding the Transpose
The transpose of a matrix involves swapping its rows and columns.
**Transpose Representation:**
\[
\begin{bmatrix}
1 & 1 \\
1 & 2
\end{bmatrix}^f
\quad becomes \quad
\begin{bmatrix}
1 & 1 \\
1 & 2
\end{bmatrix}
\]
Here, the matrix and its transpose remain the same as it’s a symmetric structure.
---
This setup helps clarify how linear constraints and objective functions are organized in matrix form, which is fundamental in linear programming and optimization.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F410204a0-8764-4473-a9c2-14bad68c17f5%2F67b22571-b0e9-4def-aaaa-efcbcd44d4fe%2Fof1xe5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Forming the Matrix for a Minimization Problem
In this exercise, we aim to construct the matrix associated with a given minimization problem and find its transpose. The transpose will be represented using "f".
#### Problem Statement
Minimize the function \( g = 4y_1 + 3y_2 \) subject to the constraints:
\[
\begin{align*}
y_1 + y_2 & \geq 17 \\
y_1 + 2y_2 & \geq 22 \\
y_1 & \geq 0, \\
y_2 & \geq 0
\end{align*}
\]
#### Constructing the Matrix
The constraints can be expressed in matrix form:
- The matrix form involves arranging the coefficients of the variables from the inequalities and objective function.
- The matrix to be filled should include the coefficients and right-side constants from the constraints.
**Matrix Representation:**
\[
\begin{bmatrix}
1 & 1 \\
1 & 2
\end{bmatrix}
\quad
\begin{bmatrix}
17 \\
22
\end{bmatrix}
\]
#### Finding the Transpose
The transpose of a matrix involves swapping its rows and columns.
**Transpose Representation:**
\[
\begin{bmatrix}
1 & 1 \\
1 & 2
\end{bmatrix}^f
\quad becomes \quad
\begin{bmatrix}
1 & 1 \\
1 & 2
\end{bmatrix}
\]
Here, the matrix and its transpose remain the same as it’s a symmetric structure.
---
This setup helps clarify how linear constraints and objective functions are organized in matrix form, which is fundamental in linear programming and optimization.
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