Finding Eigenvalues and Eigenvectors of a 3x3 Matrix

Math Problem Statement

Find the eigenvalues and eigenvectors of the matrix: A = [[6, -1, 2], [2, 3, -1], [7, 6, -4]]

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It seems to ask for the eigenvalues and eigenvectors of the given matrix. The matrix appears to be:

6 & -1 & 2 \\ 2 & 3 & -1 \\ 7 & 6 & -4 \end{bmatrix}$$ I'll solve for the eigenvalues and eigenvectors of this matrix.The eigenvalues of the matrix $$A$$ are approximately: $$\lambda_1 = 7.262, \quad \lambda_2 = -4.106, \quad \lambda_3 = 1.844$$ The corresponding eigenvectors are: $$v_1 = \begin{bmatrix} -0.7709 \\ -0.2216 \\ -0.5972 \end{bmatrix}, \quad v_2 = \begin{bmatrix} -0.1732 \\ 0.1849 \\ 0.9674 \end{bmatrix}, \quad v_3 = \begin{bmatrix} 0.1225 \\ -0.7622 \\ -0.6357 \end{bmatrix}$$ Would you like more details or further explanation? Here are some related questions for deeper understanding: 1. What are eigenvalues and eigenvectors used for in real-world applications? 2. How can you confirm that the computed vectors are indeed eigenvectors? 3. How do eigenvalues relate to the determinant of a matrix? 4. What is the geometric interpretation of eigenvalues and eigenvectors? 5. How does the characteristic polynomial help in finding eigenvalues? **Tip:** When solving for eigenvalues, ensure the matrix is square, as only square matrices have eigenvalues.

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Linear Algebra
Eigenvalues
Eigenvectors

Determinant of a matrix
Characteristic equation: det(A - λI) = 0

Eigenvalue theorem
Eigenvector definition

College Level

Eigenvectors and Eigenvalues of a 3x3 Matrix: Step-by-Step Solution

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Find Eigenvalues and Eigenvectors of a 3x3 Matrix