Find the matrix A of the quadratic form associated with the equation. 3x2 - 8xy-3y2 + 9 = 0 A = λ = X Find the eigenvalues of A. (Enter your answers as a comma-separated list.) X P = Find an orthogonal matrix P such that PTAP is diagonal. (Enter the matrix in the form [[row 1], [row 2], ...], where each row is a comma-separated list.)
Find the matrix A of the quadratic form associated with the equation. 3x2 - 8xy-3y2 + 9 = 0 A = λ = X Find the eigenvalues of A. (Enter your answers as a comma-separated list.) X P = Find an orthogonal matrix P such that PTAP is diagonal. (Enter the matrix in the form [[row 1], [row 2], ...], where each row is a comma-separated list.)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Quadratic Forms and Matrix Representation
In this exercise, you are required to find the matrix \(A\) associated with the given quadratic equation, determine the eigenvalues of \(A\), and find an orthogonal matrix \(P\) that diagonalizes \(A\).
#### Given Quadratic Equation:
\[3x^2 - 8xy - 3y^2 + 9 = 0\]
#### Tasks:
1. **Find the matrix \(A\) of the quadratic form associated with the equation.**
\[
A = \begin{pmatrix}
\boxed{\ \ \ } & \boxed{\ \ \ } \\
\boxed{\ \ \ } & \boxed{\ \ \ }
\end{pmatrix}
\]
2. **Find the eigenvalues of \(A\).** (Enter your answers as a comma-separated list.)
\[
\lambda = \boxed{\ \ \ }
\]
3. **Find an orthogonal matrix \(P\) such that \(P^TAP\) is diagonal.** (Enter the matrix in the form \([[\text{row 1}], [\text{row 2}], \dots]\), where each row is a comma-separated list.)
\[
P = \boxed{\ \ \ }
\]
**Explanation of Steps:**
1. **Construct Matrix \(A\)**:
The given quadratic equation can be written in matrix form \( \mathbf{x}^T A \mathbf{x} \), where \( \mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix} \) and \( A \) is the symmetric matrix representing the quadratic form. Here, we match the coefficients of the quadratic equation to determine \(A\).
2. **Eigenvalues of \(A\)**:
To find the eigenvalues of \(A\), solve the characteristic equation \(\det(A - \lambda I) = 0\), where \(I\) is the identity matrix. The solutions \(\lambda\) are the eigenvalues.
3. **Orthogonal Matrix \(P\)**:
The orthogonal matrix \(P\) that diagonalizes \(A\) can be found using the eigenvectors of \(A\). Matrix \(P\) is constructed by using normalized eigenvectors as its columns.
Make sure to fill in the boxes with your calculations](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03210149-7bd1-493b-9cd1-96815f185546%2F83fc628e-4175-4bf7-a136-dbe2a62817ce%2Fr1s1bgl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Quadratic Forms and Matrix Representation
In this exercise, you are required to find the matrix \(A\) associated with the given quadratic equation, determine the eigenvalues of \(A\), and find an orthogonal matrix \(P\) that diagonalizes \(A\).
#### Given Quadratic Equation:
\[3x^2 - 8xy - 3y^2 + 9 = 0\]
#### Tasks:
1. **Find the matrix \(A\) of the quadratic form associated with the equation.**
\[
A = \begin{pmatrix}
\boxed{\ \ \ } & \boxed{\ \ \ } \\
\boxed{\ \ \ } & \boxed{\ \ \ }
\end{pmatrix}
\]
2. **Find the eigenvalues of \(A\).** (Enter your answers as a comma-separated list.)
\[
\lambda = \boxed{\ \ \ }
\]
3. **Find an orthogonal matrix \(P\) such that \(P^TAP\) is diagonal.** (Enter the matrix in the form \([[\text{row 1}], [\text{row 2}], \dots]\), where each row is a comma-separated list.)
\[
P = \boxed{\ \ \ }
\]
**Explanation of Steps:**
1. **Construct Matrix \(A\)**:
The given quadratic equation can be written in matrix form \( \mathbf{x}^T A \mathbf{x} \), where \( \mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix} \) and \( A \) is the symmetric matrix representing the quadratic form. Here, we match the coefficients of the quadratic equation to determine \(A\).
2. **Eigenvalues of \(A\)**:
To find the eigenvalues of \(A\), solve the characteristic equation \(\det(A - \lambda I) = 0\), where \(I\) is the identity matrix. The solutions \(\lambda\) are the eigenvalues.
3. **Orthogonal Matrix \(P\)**:
The orthogonal matrix \(P\) that diagonalizes \(A\) can be found using the eigenvectors of \(A\). Matrix \(P\) is constructed by using normalized eigenvectors as its columns.
Make sure to fill in the boxes with your calculations
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