Given A 18 = 9 -54 12 -81 18 find one nontrivial solution of Ax = 0 by inspection. 8 -27 2
Given A 18 = 9 -54 12 -81 18 find one nontrivial solution of Ax = 0 by inspection. 8 -27 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
find one nontrivial solution of AX→=0→ by inspection.
![**Problem Statement**
Given the matrix:
\[
A = \begin{bmatrix} 9 & -2 \\ -54 & 12 \\ -81 & 18 \end{bmatrix}
\]
Find one nontrivial solution of \( A\vec{x} = \vec{0} \) by inspection.
The vector \(\vec{x}\) can be represented as:
\[
\vec{x} = \begin{bmatrix} \text{[ ]} \\ \text{[ ]} \end{bmatrix}
\]
**Explanation**
To solve the problem, we need to identify a nontrivial (nonzero) vector \(\vec{x}\) such that when multiplied with matrix \(A\), it yields the zero vector. This involves understanding linear dependencies between the rows or columns of matrix \(A\) to find a vector \(\vec{x}\) that satisfies the equation \(A\vec{x} = \vec{0}\).
**Steps for Solution**
1. **Identify Linear Dependencies:** Look for proportional rows or columns as they might lead to zero products when combined with suitable linear combinations.
2. **Choose Appropriate Values for \(\vec{x}\):** Use the observed dependencies or chosen parameters to construct a nontrivial \(\vec{x}\).
3. **Verify the Solution:** Substitute the nontrivial \(\vec{x}\) back into the equation to ensure it results in the zero vector.
This setup highlights the importance of understanding matrix operations and encourages solutions through direct observation and reasoning.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18574973-f25e-4ab8-b7d6-6007b5b87fc4%2F8edc8c77-5332-4725-a7d8-72648ec19279%2Fvhqpi08_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Given the matrix:
\[
A = \begin{bmatrix} 9 & -2 \\ -54 & 12 \\ -81 & 18 \end{bmatrix}
\]
Find one nontrivial solution of \( A\vec{x} = \vec{0} \) by inspection.
The vector \(\vec{x}\) can be represented as:
\[
\vec{x} = \begin{bmatrix} \text{[ ]} \\ \text{[ ]} \end{bmatrix}
\]
**Explanation**
To solve the problem, we need to identify a nontrivial (nonzero) vector \(\vec{x}\) such that when multiplied with matrix \(A\), it yields the zero vector. This involves understanding linear dependencies between the rows or columns of matrix \(A\) to find a vector \(\vec{x}\) that satisfies the equation \(A\vec{x} = \vec{0}\).
**Steps for Solution**
1. **Identify Linear Dependencies:** Look for proportional rows or columns as they might lead to zero products when combined with suitable linear combinations.
2. **Choose Appropriate Values for \(\vec{x}\):** Use the observed dependencies or chosen parameters to construct a nontrivial \(\vec{x}\).
3. **Verify the Solution:** Substitute the nontrivial \(\vec{x}\) back into the equation to ensure it results in the zero vector.
This setup highlights the importance of understanding matrix operations and encourages solutions through direct observation and reasoning.
Expert Solution
Step 1: Non trivial solution
Here for given A matrix we have to find the non trivial solution x.
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