Math Problem Statement
Solution
To find the eigenvalues of the matrix , we need to solve the characteristic equation, which is given by:
Where is an eigenvalue, and is the identity matrix. Let's solve this step by step.
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Matrix :
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Characteristic equation: Expanding this determinant:
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Solving the quadratic equation: We can solve this quadratic equation using the quadratic formula: where , , and .
Substituting these values: So, the solutions are:
The eigenvalues of the matrix are and .
Would you like a detailed explanation of how to find the eigenvectors, or do you have any other questions?
Related Questions:
- How do you find the eigenvectors corresponding to the eigenvalues and ?
- What is the geometric interpretation of eigenvalues and eigenvectors?
- Can the eigenvalues be negative, and what would that signify?
- How do eigenvalues relate to the determinant and trace of a matrix?
- What are the implications if a matrix has repeated eigenvalues?
Tip: Eigenvalues provide crucial insights into the properties of a matrix, especially in applications involving stability and oscillation in systems of differential equations.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Determinants
Formulas
Characteristic Equation: det(B - λI) = 0
Quadratic Formula: λ = (-b ± √(b² - 4ac)) / 2a
Theorems
Eigenvalue Theorem
Suitable Grade Level
Undergraduate