Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix.
Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix.
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
![**Educational Content: Parametric Vector Form Solutions**
**Task:**
Describe all solutions of \( Ax = 0 \) in parametric vector form, where \( A \) is row equivalent to the given matrix.
**Matrix:**
\[
\begin{bmatrix}
3 & -6 & 9 \\
-1 & 2 & -3
\end{bmatrix}
\]
**Solution Representation:**
The solution \( x \) is expressed as:
\[
x = x_2
\begin{bmatrix}
\Box \\
\Box \\
\Box
\end{bmatrix}
+
x_3
\begin{bmatrix}
\Box \\
\Box \\
\Box
\end{bmatrix}
\]
*(Type an integer or fraction for each matrix element.)*
**Explanation:**
- The matrix shown is a \(2 \times 3\) matrix, which means it represents a system of two linear equations with three variables.
- The task is to express the solution set of these equations in terms of free variables \( x_2 \) and \( x_3 \).
- The parametric vector form will involve expressing the solution vector \( x \) as a linear combination of the free variables multiplied by their respective vectors. Filling in the boxes will provide a complete description of the solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5362cbea-9427-4b2b-99cb-57a208edd853%2F82dab2e7-2dd8-414b-b998-a2fe3fda26af%2Fdgzqjmh_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Content: Parametric Vector Form Solutions**
**Task:**
Describe all solutions of \( Ax = 0 \) in parametric vector form, where \( A \) is row equivalent to the given matrix.
**Matrix:**
\[
\begin{bmatrix}
3 & -6 & 9 \\
-1 & 2 & -3
\end{bmatrix}
\]
**Solution Representation:**
The solution \( x \) is expressed as:
\[
x = x_2
\begin{bmatrix}
\Box \\
\Box \\
\Box
\end{bmatrix}
+
x_3
\begin{bmatrix}
\Box \\
\Box \\
\Box
\end{bmatrix}
\]
*(Type an integer or fraction for each matrix element.)*
**Explanation:**
- The matrix shown is a \(2 \times 3\) matrix, which means it represents a system of two linear equations with three variables.
- The task is to express the solution set of these equations in terms of free variables \( x_2 \) and \( x_3 \).
- The parametric vector form will involve expressing the solution vector \( x \) as a linear combination of the free variables multiplied by their respective vectors. Filling in the boxes will provide a complete description of the solutions.
![**Matrix Problem Description**
Consider the task to describe all solutions of \( \mathbf{Ax} = \mathbf{0} \) in parametric vector form, where \(\mathbf{A}\) is row equivalent to the given matrix:
\[
\begin{bmatrix}
1 & 5 & -3 & -1 & 0 & 6 \\
0 & 0 & 1 & 0 & 0 & 4 \\
0 & 0 & 0 & 1 & 0 & -4 \\
0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]
**Solution Vector Formulation**
We express the solution vector \( \mathbf{x} \) in terms of free variables \( x_2, x_5, \) and \( x_6 \):
\[
\mathbf{x} = x_2 \begin{bmatrix} ? \\ ? \\ ? \\ ? \\ ? \\ ? \end{bmatrix} + x_5 \begin{bmatrix} ? \\ ? \\ ? \\ ? \\ ? \\ ? \end{bmatrix} + x_6 \begin{bmatrix} ? \\ ? \\ ? \\ ? \\ ? \\ ? \end{bmatrix}
\]
*(Type an integer or fraction for each matrix element.)*
This setup initiates the process to find the parametric vector form solution where \( x_2, x_5, \) and \( x_6 \) are the parameters. These steps require solving the matrix to determine each vector component's contribution from the free variables.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5362cbea-9427-4b2b-99cb-57a208edd853%2F82dab2e7-2dd8-414b-b998-a2fe3fda26af%2Fvc62le3_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Problem Description**
Consider the task to describe all solutions of \( \mathbf{Ax} = \mathbf{0} \) in parametric vector form, where \(\mathbf{A}\) is row equivalent to the given matrix:
\[
\begin{bmatrix}
1 & 5 & -3 & -1 & 0 & 6 \\
0 & 0 & 1 & 0 & 0 & 4 \\
0 & 0 & 0 & 1 & 0 & -4 \\
0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]
**Solution Vector Formulation**
We express the solution vector \( \mathbf{x} \) in terms of free variables \( x_2, x_5, \) and \( x_6 \):
\[
\mathbf{x} = x_2 \begin{bmatrix} ? \\ ? \\ ? \\ ? \\ ? \\ ? \end{bmatrix} + x_5 \begin{bmatrix} ? \\ ? \\ ? \\ ? \\ ? \\ ? \end{bmatrix} + x_6 \begin{bmatrix} ? \\ ? \\ ? \\ ? \\ ? \\ ? \end{bmatrix}
\]
*(Type an integer or fraction for each matrix element.)*
This setup initiates the process to find the parametric vector form solution where \( x_2, x_5, \) and \( x_6 \) are the parameters. These steps require solving the matrix to determine each vector component's contribution from the free variables.
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