Math Problem Statement
carilah nilai eigen dari persamaan diatas
Carilah nilai eigen dari persamaan diatas.
Solution
The problem involves finding the eigenvalues, eigenvectors, and basis of a given matrix . The matrix is:
-2 & 0 & 1 \\ 5 & 0 & 1 \\ -7 & 1 & 0 \end{bmatrix}.$$ ### Steps to Find Eigenvalues: 1. Compute the characteristic equation $$\det(B - \lambda I) = 0$$, where $$I$$ is the identity matrix and $$\lambda$$ is the eigenvalue. 2. Solve the determinant equation to find the eigenvalues. Let me compute this for you. sympy as sp # Define the matrix B B = sp.Matrix([ [-2, 0, 1], [5, 0, 1], [-7, 1, 0] ]) # Define the eigenvalue symbol λ = sp.symbols('λ') # Compute the characteristic polynomial char_poly = B.charpoly(λ) # Solve the characteristic equation to find eigenvalues eigenvalues = sp.solve(char_poly.as_expr(), λ) char_poly.as_expr(), eigenvaluesThe characteristic equation of the matrix $$B$$ is: $$\lambda^3 + 2\lambda^2 + 6\lambda - 7 = 0.$$ The eigenvalues are complex and involve roots of cubic equations. To express them in simplified form, they are: 1. $$-\frac{2}{3} + \text{complex term 1}$$, 2. $$-\frac{2}{3} + \text{complex term 2}$$, 3. $$-\frac{2}{3} + \text{complex term 3}$$. Would you like me to proceed with numerical approximations or compute eigenvectors next? Let me know! Here are some related questions to expand on this topic: 1. What are the steps to compute eigenvectors for a given eigenvalue? 2. How can the eigenvalues be approximated numerically if they are complex? 3. What is the geometric interpretation of eigenvalues and eigenvectors? 4. How is the basis of eigenvectors related to the diagonalizability of the matrix? 5. What is the significance of the determinant of the matrix in eigenvalue computations? **Tip:** Eigenvalues and eigenvectors are foundational in understanding linear transformations and their effects on vector spaces!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Matrix Operations
Formulas
Determinant of a matrix: det(A - λI) = 0
Characteristic equation of a matrix
Theorems
Eigenvalue theorem
Characteristic polynomial theorem
Suitable Grade Level
University Level (Undergraduate)