Determine if the system has a nontrivial solution. Try to use as few row operations as possible. - 3x, + 4x2 - 8x3 = 0 - 12x, + 6x, + 4x3 = 0
Determine if the system has a nontrivial solution. Try to use as few row operations as possible. - 3x, + 4x2 - 8x3 = 0 - 12x, + 6x, + 4x3 = 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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![### Linear Algebra: Nontrivial Solutions of Systems of Equations
#### Problem Statement:
Determine if the system of equations has a nontrivial solution. Try to use as few row operations as possible.
\[
-3x_1 + 4x_2 - 8x_3 = 0
\]
\[
-12x_1 + 6x_2 + 4x_3 = 0
\]
#### Question:
Choose the correct answer below:
- **A.** The system has only a trivial solution.
- **B.** The system has a nontrivial solution.
- **C.** It is impossible to determine.
#### Explanation:
To determine if this system of equations has a nontrivial solution, we look at the possibility of the system having more solutions than just the trivial solution \( x_1 = x_2 = x_3 = 0 \). For this, we use methods from linear algebra, such as row reduction, to examine the system.
Let's outline the process:
1. **Form the Augmented Matrix:**
\[
\begin{pmatrix}
-3 & 4 & -8 & | & 0 \\
-12 & 6 & 4 & | & 0
\end{pmatrix}
\]
2. **Perform Row Operations to Simplify:**
- Divide the first row by -3:
\[
\begin{pmatrix}
1 & -\frac{4}{3} & \frac{8}{3} & | & 0 \\
-12 & 6 & 4 & | & 0
\end{pmatrix}
\]
- Use this result to eliminate the first entry of the second row.
\[
\begin{pmatrix}
1 & -\frac{4}{3} & \frac{8}{3} & | & 0 \\
0 & 0 & -\frac{4}{3} & | & 0
\end{pmatrix}
\]
From the resulting matrix, we see that there are fewer pivots than variables, indicating that the system is consistent and has free variables, implying nontrivial solutions exist.
Thus, the correct choice is:
- **B. The system has a nontrivial solution.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0867650-b445-489d-98f3-e27e64467d14%2Fafc9c927-0d42-49fe-a89a-b40100809ec9%2Fxuuj2yi_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Algebra: Nontrivial Solutions of Systems of Equations
#### Problem Statement:
Determine if the system of equations has a nontrivial solution. Try to use as few row operations as possible.
\[
-3x_1 + 4x_2 - 8x_3 = 0
\]
\[
-12x_1 + 6x_2 + 4x_3 = 0
\]
#### Question:
Choose the correct answer below:
- **A.** The system has only a trivial solution.
- **B.** The system has a nontrivial solution.
- **C.** It is impossible to determine.
#### Explanation:
To determine if this system of equations has a nontrivial solution, we look at the possibility of the system having more solutions than just the trivial solution \( x_1 = x_2 = x_3 = 0 \). For this, we use methods from linear algebra, such as row reduction, to examine the system.
Let's outline the process:
1. **Form the Augmented Matrix:**
\[
\begin{pmatrix}
-3 & 4 & -8 & | & 0 \\
-12 & 6 & 4 & | & 0
\end{pmatrix}
\]
2. **Perform Row Operations to Simplify:**
- Divide the first row by -3:
\[
\begin{pmatrix}
1 & -\frac{4}{3} & \frac{8}{3} & | & 0 \\
-12 & 6 & 4 & | & 0
\end{pmatrix}
\]
- Use this result to eliminate the first entry of the second row.
\[
\begin{pmatrix}
1 & -\frac{4}{3} & \frac{8}{3} & | & 0 \\
0 & 0 & -\frac{4}{3} & | & 0
\end{pmatrix}
\]
From the resulting matrix, we see that there are fewer pivots than variables, indicating that the system is consistent and has free variables, implying nontrivial solutions exist.
Thus, the correct choice is:
- **B. The system has a nontrivial solution.**
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