Is the vector [1/2] 1 in the null space of matrix A 8 10 L22 7 3 -1 13 -6? 2
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
![**Problem Statement:**
Is the vector \( \begin{bmatrix} \frac{1}{2} \\ -1 \\ 1 \end{bmatrix} \) in the null space of matrix \( A = \begin{bmatrix} 8 & 7 & 3 \\ 10 & -1 & -6 \\ 22 & 13 & 2 \end{bmatrix} \)?
**Explanation:**
To determine if the vector \( \begin{bmatrix} \frac{1}{2} \\ -1 \\ 1 \end{bmatrix} \) is in the null space of matrix \( A \), we must check if the following matrix-vector multiplication results in the zero vector:
\[ A \cdot \begin{bmatrix} \frac{1}{2} \\ -1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \]
If this condition holds true, the vector is in the null space of matrix \( A \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2432691a-c29a-4a02-b85a-93b716867369%2Fc244e62f-6a37-42f9-a46e-5cca6ff82b2a%2Fpbfw1i4_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Is the vector \( \begin{bmatrix} \frac{1}{2} \\ -1 \\ 1 \end{bmatrix} \) in the null space of matrix \( A = \begin{bmatrix} 8 & 7 & 3 \\ 10 & -1 & -6 \\ 22 & 13 & 2 \end{bmatrix} \)?
**Explanation:**
To determine if the vector \( \begin{bmatrix} \frac{1}{2} \\ -1 \\ 1 \end{bmatrix} \) is in the null space of matrix \( A \), we must check if the following matrix-vector multiplication results in the zero vector:
\[ A \cdot \begin{bmatrix} \frac{1}{2} \\ -1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \]
If this condition holds true, the vector is in the null space of matrix \( A \).
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Step 1: Explanation
Let us check whether the given vector is in the null space of given matrix A.
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