Let Describe all solutions of Ax = 0. 18 = x2 X 1 -7 -6 0 + x3 X 5 A = -35 -30 0 [ 1 5 -7 -35 +x4 :] -6 0 30 0 00
Let Describe all solutions of Ax = 0. 18 = x2 X 1 -7 -6 0 + x3 X 5 A = -35 -30 0 [ 1 5 -7 -35 +x4 :] -6 0 30 0 00
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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![# Matrix-Vector Multiplication
## Description
**Topic:** Matrix-Vector Multiplication
**Score:** 12/18 (13/18 answered)
### Problem Statement
Given the matrix \( A \):
\[
A = \begin{bmatrix} 1 & -7 & -6 & 0 \\ 5 & -35 & -30 & 0 \end{bmatrix}
\]
Describe all solutions of the homogeneous system \( A\vec{x} = \vec{0} \).
### Solution Structure
The general solution to the system is expressed in terms of free variables \( x_2 \), \( x_3 \), and \( x_4 \):
\[
\vec{x} = x_2 \begin{bmatrix} 1 \\ -7 \\ -6 \\ 0 \end{bmatrix} + x_3 \begin{bmatrix} 5 \\ -35 \\ -30 \\ 0 \end{bmatrix} + x_4 \begin{bmatrix} \\ \\ \\ \end{bmatrix}
\]
### Explanation
- **Coefficient Matrix \( A \):** A 2x4 matrix, indicating two equations and four variables.
- **Solution Format:** The solution is represented as a linear combination of vectors multiplied by the free variables \( x_2 \), \( x_3 \), and \( x_4 \).
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Transcribed Image Text:# Matrix-Vector Multiplication
## Description
**Topic:** Matrix-Vector Multiplication
**Score:** 12/18 (13/18 answered)
### Problem Statement
Given the matrix \( A \):
\[
A = \begin{bmatrix} 1 & -7 & -6 & 0 \\ 5 & -35 & -30 & 0 \end{bmatrix}
\]
Describe all solutions of the homogeneous system \( A\vec{x} = \vec{0} \).
### Solution Structure
The general solution to the system is expressed in terms of free variables \( x_2 \), \( x_3 \), and \( x_4 \):
\[
\vec{x} = x_2 \begin{bmatrix} 1 \\ -7 \\ -6 \\ 0 \end{bmatrix} + x_3 \begin{bmatrix} 5 \\ -35 \\ -30 \\ 0 \end{bmatrix} + x_4 \begin{bmatrix} \\ \\ \\ \end{bmatrix}
\]
### Explanation
- **Coefficient Matrix \( A \):** A 2x4 matrix, indicating two equations and four variables.
- **Solution Format:** The solution is represented as a linear combination of vectors multiplied by the free variables \( x_2 \), \( x_3 \), and \( x_4 \).
### Interaction
You can interact with the question by selecting "Next question," "Get a similar question," or retry this question for better understanding.
### Submission
Press the "Submit Question" button to finalize your answer.
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