Math Problem Statement
Compute the eigenvalues of the matrix [-3, -3; 6, 3]. Enter the values a and b such that your answer is in the form λ = a ± bi.
Solution
To compute the eigenvalues of the matrix:
-3 & -3 \\ 6 & 3 \end{bmatrix}$$ We need to find the values of $$\lambda$$ that satisfy the characteristic equation: $$\det(A - \lambda I) = 0$$ Where $$A$$ is the given matrix, and $$I$$ is the identity matrix. This gives us: $$\det\left(\begin{bmatrix} -3 & -3 \\ 6 & 3 \end{bmatrix} - \lambda \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\right) = 0$$ This simplifies to: $$\det\begin{bmatrix} -3 - \lambda & -3 \\ 6 & 3 - \lambda \end{bmatrix} = 0$$ Now, calculate the determinant: $$(-3 - \lambda)(3 - \lambda) - (-3)(6) = 0$$ Expanding this: $$(-3 - \lambda)(3 - \lambda) + 18 = 0$$ $$(-9 + 3\lambda - 3\lambda - \lambda^2) + 18 = 0$$ $$-\lambda^2 + 9 = 0$$ $$\lambda^2 = 9$$ $$\lambda = \pm 3i$$ So, the eigenvalues are $$\lambda = 0 \pm 3i$$, meaning $$a = 0$$ and $$b = 3$$. Would you like further details or clarifications on this? Here are five related questions to expand your understanding: 1. How do we find eigenvectors once we have eigenvalues? 2. What does it mean when eigenvalues are purely imaginary? 3. Can a non-square matrix have eigenvalues? 4. How does the trace of a matrix relate to its eigenvalues? 5. What is the geometric interpretation of complex eigenvalues? **Tip:** For 2x2 matrices, the characteristic equation is always a quadratic, making it easier to solve for eigenvalues.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Matrices
Complex Numbers
Formulas
Determinant formula: det(A - λI) = 0
Quadratic equation from determinant
Theorems
Eigenvalue Theorem
Quadratic Formula
Suitable Grade Level
Grades 11-12, University