Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix. 1 - 5 -1 0 -2 7 1 0 0 3 0 0 1 5 0 0 0 0 x=x2+ x4+xeO + X6 (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. 1 - 5 -1 0 -2 7 1 0 0 3 0 0 1 5 0 0 0 0 x=x2+ x4+xeO + X6 (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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![### Educational Content: Solving Homogeneous Systems in Parametric Vector Form
In this section, we explore how to describe all solutions of the equation \( Ax = 0 \) in parametric vector form, where matrix \( A \) is row equivalent to the matrix provided below:
\[
\begin{bmatrix}
1 & -5 & -1 & 0 & -2 & 7 \\
0 & 0 & 1 & 0 & 0 & 3 \\
0 & 0 & 0 & 0 & 1 & 5 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\end{bmatrix}
\]
The task is to express the solution vector \( x \) in terms of the free variables \( x_2, x_4,\) and \( x_6 \).
### Solution:
The solution vector \( x \) can be described as:
\[
x = x_2
\begin{bmatrix}
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\end{bmatrix}
+ x_4
\begin{bmatrix}
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\end{bmatrix}
+ x_6
\begin{bmatrix}
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\end{bmatrix}
\]
Complete the solution by filling in each blank with the appropriate integer or fraction as required.
*Note: The solution encourages students to analyze the row-reduced form of the matrix and recognize the leading and free variables to express the solution in parametric form.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d31c5b4-c6dc-4739-be04-6ff777f4b60f%2Ff44433ea-f481-45a6-919e-d8114e1c681b%2Fg3jkl1q_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Content: Solving Homogeneous Systems in Parametric Vector Form
In this section, we explore how to describe all solutions of the equation \( Ax = 0 \) in parametric vector form, where matrix \( A \) is row equivalent to the matrix provided below:
\[
\begin{bmatrix}
1 & -5 & -1 & 0 & -2 & 7 \\
0 & 0 & 1 & 0 & 0 & 3 \\
0 & 0 & 0 & 0 & 1 & 5 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\end{bmatrix}
\]
The task is to express the solution vector \( x \) in terms of the free variables \( x_2, x_4,\) and \( x_6 \).
### Solution:
The solution vector \( x \) can be described as:
\[
x = x_2
\begin{bmatrix}
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\end{bmatrix}
+ x_4
\begin{bmatrix}
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\end{bmatrix}
+ x_6
\begin{bmatrix}
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\text{blank} \\
\end{bmatrix}
\]
Complete the solution by filling in each blank with the appropriate integer or fraction as required.
*Note: The solution encourages students to analyze the row-reduced form of the matrix and recognize the leading and free variables to express the solution in parametric form.*
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