Math Problem Statement
We start by solving the eigenvalue problem for 𝐴 A, which involves solving the characteristic equation:
det ( 𝐴 − 𝜆 𝐼 )
0 det(A−λI)=0 where 𝜆 λ is the eigenvalue and 𝐼 I is the identity matrix.
𝐴 − 𝜆 𝐼
( 0 − 𝜆 1 − 2 − 3 − 𝜆 ) A−λI=( 0−λ −2
1 −3−λ ) det ( 𝐴 − 𝜆 𝐼 )
det ( − 𝜆 1 − 2 − 3 − 𝜆 ) det(A−λI)=det( −λ −2
1 −3−λ )
Solution
To solve the eigenvalue problem for matrix , we need to compute the determinant of the matrix , where is given as:
The identity matrix is:
Step 1: Construct
Now, subtract from matrix :
= \begin{pmatrix} 0 - \lambda & 1 \\ -2 & -3 - \lambda \end{pmatrix} = \begin{pmatrix} -\lambda & 1 \\ -2 & -3 - \lambda \end{pmatrix}$$ ### Step 2: Compute the determinant The determinant of the matrix $$A - \lambda I$$ is: $$\text{det}(A - \lambda I) = \text{det} \begin{pmatrix} -\lambda & 1 \\ -2 & -3 - \lambda \end{pmatrix}$$ Using the determinant formula for a 2x2 matrix $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$, where the determinant is $$ad - bc$$, we calculate: $$\text{det}(A - \lambda I) = (-\lambda)(-3 - \lambda) - (1)(-2)$$ $$= \lambda(3 + \lambda) + 2$$ $$= \lambda^2 + 3\lambda + 2$$ ### Step 3: Solve the characteristic equation Set the determinant equal to zero to solve for the eigenvalues $$\lambda$$: $$\lambda^2 + 3\lambda + 2 = 0$$ Factor the quadratic equation: $$(\lambda + 1)(\lambda + 2) = 0$$ Thus, the eigenvalues are: $$\lambda_1 = -1 \quad \text{and} \quad \lambda_2 = -2$$ ### Conclusion: The eigenvalues of the matrix $$A$$ are $$\lambda_1 = -1$$ and $$\lambda_2 = -2$$. Do you want to see the steps for finding the eigenvectors or have any other questions? Here are some related questions: 1. What are the eigenvectors corresponding to each eigenvalue? 2. How do we interpret the eigenvalues in terms of matrix transformations? 3. What are some applications of eigenvalues and eigenvectors in real-world problems? 4. Can a matrix have complex eigenvalues, and under what conditions? 5. How do diagonalization and eigenvalues relate to each other? **Tip**: Always check if the matrix is symmetric, as symmetric matrices have real eigenvalues.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Determinants
Matrices
Formulas
det(A - λI) = 0
Determinant of 2x2 matrix: det(A) = ad - bc
Quadratic equation: λ^2 + 3λ + 2 = 0
Theorems
Eigenvalue theorem
Determinant formula for 2x2 matrices
Suitable Grade Level
Undergraduate level (Grades 11-12 or college level)