Math Problem Statement
Find the eigenvalues and eigenvectors of the matrix:
\[ A = \begin{pmatrix} 8 & -6 & 2 \ -6 & 7 & -4 \ 2 & -4 & 3 \end{pmatrix} \]
Solution
The problem in the image asks to find the eigenvalues and eigenvectors of the matrix:
8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{pmatrix}$$ ### Step 1: Find the eigenvalues The eigenvalues ($$\lambda$$) are found by solving the characteristic equation: $$\text{det}(A - \lambda I) = 0$$ Where $$I$$ is the identity matrix, and $$\lambda$$ represents the eigenvalues. $$A - \lambda I = \begin{pmatrix} 8 - \lambda & -6 & 2 \\ -6 & 7 - \lambda & -4 \\ 2 & -4 & 3 - \lambda \end{pmatrix}$$ We compute the determinant of this matrix and set it equal to zero to find the eigenvalues. ### Step 2: Find the eigenvectors After determining the eigenvalues, the corresponding eigenvectors are found by solving: $$(A - \lambda I) \vec{v} = 0$$ Where $$\vec{v}$$ is the eigenvector corresponding to the eigenvalue $$\lambda$$. Let me proceed with the computation of the eigenvalues and eigenvectors. numpy as np # Defining the matrix A A = np.array([[8, -6, 2], [-6, 7, -4], [2, -4, 3]]) # Calculating the eigenvalues and eigenvectors eigenvalues, eigenvectors = np.linalg.eig(A) eigenvalues, eigenvectorsThe eigenvalues of the matrix are: $$\lambda_1 = 15, \quad \lambda_2 = 3, \quad \lambda_3 \approx 0$$ The corresponding eigenvectors are: - For $$\lambda_1 = 15$$, the eigenvector is $$\begin{pmatrix} -\frac{2}{3} \\ \frac{2}{3} \\ -\frac{1}{3} \end{pmatrix}$$ - For $$\lambda_2 = 3$$, the eigenvector is $$\begin{pmatrix} \frac{2}{3} \\ \frac{1}{3} \\ -\frac{2}{3} \end{pmatrix}$$ - For $$\lambda_3 \approx 0$$, the eigenvector is $$\begin{pmatrix} \frac{1}{3} \\ \frac{2}{3} \\ \frac{2}{3} \end{pmatrix}$$ Let me know if you need more details or explanations! Here are 5 related questions you might explore: 1. How do we interpret the significance of eigenvalues and eigenvectors in different applications? 2. Can you solve for the eigenvectors of a 2x2 matrix? 3. How does the presence of a zero eigenvalue affect the matrix? 4. What are the practical uses of eigenvalues in physics or engineering? 5. How do you compute the determinant of a 3x3 matrix? **Tip**: If a matrix has an eigenvalue of zero, it indicates that the matrix is singular and not invertible.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrices
Determinants
Formulas
det(A - \lambda I) = 0
(A - \lambda I) v = 0
Theorems
Eigenvalue Theorem
Characteristic Equation
Suitable Grade Level
Grades 10-12