A = II 18x28xy + 18y² - 74 = 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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![**Title: Finding the Matrix of a Quadratic Form**
**Objective:**
Learn how to find the matrix associated with a given quadratic form.
**Problem Statement:**
Find the matrix of the quadratic form associated with the equation:
\[ 18x^2 - 8xy + 18y^2 - 74 = 0 \]
**Matrix Representation:**
The quadratic form can be expressed in matrix notation as \( \mathbf{x}^T A \mathbf{x} \), where \( \mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix} \).
**Matrix \( A \):**
\[ A = \begin{bmatrix} a & b/2 \\ b/2 & c \end{bmatrix} \]
For the quadratic function \( ax^2 + bxy + cy^2 \), the coefficients are:
- \( a = 18 \)
- \( b = -8 \)
- \( c = 18 \)
Thus, the matrix \( A \) is:
\[ A = \begin{bmatrix} 18 & -4 \\ -4 & 18 \end{bmatrix} \]
**Summary:**
The matrix \( A \) corresponding to the given quadratic form is a symmetric matrix, which plays a crucial role in various applications like analyzing conic sections and performing transformations.
Understanding this representation is essential for simplifying and solving systems involving quadratic forms in mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61ff295b-afd4-43c3-8ed2-2cc17b9c2249%2F86483f6c-70bf-42cc-b279-b767e1735d66%2Fr7y2i1o_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Finding the Matrix of a Quadratic Form**
**Objective:**
Learn how to find the matrix associated with a given quadratic form.
**Problem Statement:**
Find the matrix of the quadratic form associated with the equation:
\[ 18x^2 - 8xy + 18y^2 - 74 = 0 \]
**Matrix Representation:**
The quadratic form can be expressed in matrix notation as \( \mathbf{x}^T A \mathbf{x} \), where \( \mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix} \).
**Matrix \( A \):**
\[ A = \begin{bmatrix} a & b/2 \\ b/2 & c \end{bmatrix} \]
For the quadratic function \( ax^2 + bxy + cy^2 \), the coefficients are:
- \( a = 18 \)
- \( b = -8 \)
- \( c = 18 \)
Thus, the matrix \( A \) is:
\[ A = \begin{bmatrix} 18 & -4 \\ -4 & 18 \end{bmatrix} \]
**Summary:**
The matrix \( A \) corresponding to the given quadratic form is a symmetric matrix, which plays a crucial role in various applications like analyzing conic sections and performing transformations.
Understanding this representation is essential for simplifying and solving systems involving quadratic forms in mathematics.
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