Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. 3 - - 1 -9 12 3 - 4 X=X₂ +X3 (Type an integer or fraction for each matrix element.)
Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. 3 - - 1 -9 12 3 - 4 X=X₂ +X3 (Type an integer or fraction for each matrix element.)
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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![### Solving Ax = 0 in Parametric Vector Form
To describe all solutions of the homogeneous equation \( \mathbf{A}\mathbf{x} = \mathbf{0} \), we need to express the solutions in parametric vector form. Given that matrix \( \mathbf{A} \) is row equivalent to the following matrix:
\[ \mathbf{A} = \begin{bmatrix} 3 & -9 & 12 \\ -1 & 3 & -4 \end{bmatrix} \]
Let's find the solution \( \mathbf{x} \) as a vector dependent on free variables \( x_2 \) and \( x_3 \).
1. **Matrix Representation:**
\[
\mathbf{A} =
\begin{bmatrix}
3 & -9 & 12 \\
-1 & 3 & -4
\end{bmatrix}
\]
2. **Expressing the Solutions:**
The parametric solution \( \mathbf{x} \) can be written as:
\[
\mathbf{x} = x_2 \begin{bmatrix} \_ \\ \_ \\ \_ \end{bmatrix} + x_3 \begin{bmatrix} \_ \\ \_ \\ \_ \end{bmatrix}
\]
3. **Typing the Elements:**
Each element of the matrices should be an integer or fraction.
### Example Solution:
If you simplify you might get a result in the form:
\[
\mathbf{x} = x_2 \begin{bmatrix} x_{21} \\ x_{22} \\ x_{23} \end{bmatrix} + x_3 \begin{bmatrix} x_{31} \\ x_{32} \\ x_{33} \end{bmatrix}
\]
Here, \( x_{21} \), \( x_{22} \), \( x_{23} \) and \( x_{31} \), \( x_{32} \), \( x_{33} \) would be values determined by performing row reduction on matrix \( \mathbf{A} \) and solving for the parametric vector form.
**Note for Typing:**
Enter integers or fractions for each matrix element where appropriate.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70fe4f99-9e69-4bef-afe6-80d54e8903ee%2F2cf09e35-5967-4010-8ad4-57b0c8922645%2Fi61dgnb_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Ax = 0 in Parametric Vector Form
To describe all solutions of the homogeneous equation \( \mathbf{A}\mathbf{x} = \mathbf{0} \), we need to express the solutions in parametric vector form. Given that matrix \( \mathbf{A} \) is row equivalent to the following matrix:
\[ \mathbf{A} = \begin{bmatrix} 3 & -9 & 12 \\ -1 & 3 & -4 \end{bmatrix} \]
Let's find the solution \( \mathbf{x} \) as a vector dependent on free variables \( x_2 \) and \( x_3 \).
1. **Matrix Representation:**
\[
\mathbf{A} =
\begin{bmatrix}
3 & -9 & 12 \\
-1 & 3 & -4
\end{bmatrix}
\]
2. **Expressing the Solutions:**
The parametric solution \( \mathbf{x} \) can be written as:
\[
\mathbf{x} = x_2 \begin{bmatrix} \_ \\ \_ \\ \_ \end{bmatrix} + x_3 \begin{bmatrix} \_ \\ \_ \\ \_ \end{bmatrix}
\]
3. **Typing the Elements:**
Each element of the matrices should be an integer or fraction.
### Example Solution:
If you simplify you might get a result in the form:
\[
\mathbf{x} = x_2 \begin{bmatrix} x_{21} \\ x_{22} \\ x_{23} \end{bmatrix} + x_3 \begin{bmatrix} x_{31} \\ x_{32} \\ x_{33} \end{bmatrix}
\]
Here, \( x_{21} \), \( x_{22} \), \( x_{23} \) and \( x_{31} \), \( x_{32} \), \( x_{33} \) would be values determined by performing row reduction on matrix \( \mathbf{A} \) and solving for the parametric vector form.
**Note for Typing:**
Enter integers or fractions for each matrix element where appropriate.
![### Solving the System of Linear Equations
#### Problem Statement:
Describe all solutions of \( A\mathbf{x} = \mathbf{0} \) in parametric vector form, where \( A \) is row equivalent to the given matrix:
\[
\begin{bmatrix}
1 & 2 & -4 & 5 \\
0 & 1 & -5 & 4
\end{bmatrix}
\]
#### Objective:
Find the parametric vector form of the solution set.
#### Steps to Solve:
1. Identify the matrix \( A \) and consider it in the row equivalent form provided.
2. Write the system of equations corresponding to the augmented matrix.
3. Solve the system by expressing the dependent variables in terms of the free variables.
4. Express the solution set in parametric vector form.
Given matrix \( A \):
\[
\begin{bmatrix}
1 & 2 & -4 & 5 \\
0 & 1 & -5 & 4
\end{bmatrix}
\]
The corresponding system of equations is:
1. \( x_1 + 2x_2 - 4x_3 + 5x_4 = 0 \)
2. \( x_2 - 5x_3 + 4x_4 = 0 \)
#### Parametric Vector Form:
\[
\mathbf{x} = x_3 \mathbf{v_1} + x_4 \mathbf{v_2}
\]
Where:
- \( x_3 \) and \( x_4 \) are free variables.
- \(\mathbf{v_1}\) and \(\mathbf{v_2}\) are vectors derived from solving the system.
The text provides incomplete vectors to be filled in:
\[
\mathbf{x} = x_3 \begin{bmatrix}
\_ \\
\_ \\
\_ \\
\_
\end{bmatrix} + x_4 \begin{bmatrix}
\_ \\
\_ \\
\_ \\
\_
\end{bmatrix}
\]
#### Instructions for Students:
1. Type an integer or fraction for each matrix element based on the row-reduced form's terms to get the specific vectors corresponding to \( \mathbf{v_1} \) and \( \mathbf{v_2} \).
2. Ensure all dependent variables are properly expressed in terms](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70fe4f99-9e69-4bef-afe6-80d54e8903ee%2F2cf09e35-5967-4010-8ad4-57b0c8922645%2Fyeip6x2_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving the System of Linear Equations
#### Problem Statement:
Describe all solutions of \( A\mathbf{x} = \mathbf{0} \) in parametric vector form, where \( A \) is row equivalent to the given matrix:
\[
\begin{bmatrix}
1 & 2 & -4 & 5 \\
0 & 1 & -5 & 4
\end{bmatrix}
\]
#### Objective:
Find the parametric vector form of the solution set.
#### Steps to Solve:
1. Identify the matrix \( A \) and consider it in the row equivalent form provided.
2. Write the system of equations corresponding to the augmented matrix.
3. Solve the system by expressing the dependent variables in terms of the free variables.
4. Express the solution set in parametric vector form.
Given matrix \( A \):
\[
\begin{bmatrix}
1 & 2 & -4 & 5 \\
0 & 1 & -5 & 4
\end{bmatrix}
\]
The corresponding system of equations is:
1. \( x_1 + 2x_2 - 4x_3 + 5x_4 = 0 \)
2. \( x_2 - 5x_3 + 4x_4 = 0 \)
#### Parametric Vector Form:
\[
\mathbf{x} = x_3 \mathbf{v_1} + x_4 \mathbf{v_2}
\]
Where:
- \( x_3 \) and \( x_4 \) are free variables.
- \(\mathbf{v_1}\) and \(\mathbf{v_2}\) are vectors derived from solving the system.
The text provides incomplete vectors to be filled in:
\[
\mathbf{x} = x_3 \begin{bmatrix}
\_ \\
\_ \\
\_ \\
\_
\end{bmatrix} + x_4 \begin{bmatrix}
\_ \\
\_ \\
\_ \\
\_
\end{bmatrix}
\]
#### Instructions for Students:
1. Type an integer or fraction for each matrix element based on the row-reduced form's terms to get the specific vectors corresponding to \( \mathbf{v_1} \) and \( \mathbf{v_2} \).
2. Ensure all dependent variables are properly expressed in terms
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