The image you've uploaded contains solutions to matrix-related problems, specifically focusing on finding eigenvalues and eigenvectors, as well as a related multiple-choice question (MCQ).

Summary of the Content:

1. Matrix Eigenvalue and Eigenvector Solutions:
   • Problem 1:
     • Matrix: Not directly shown.
     • Eigenvalues: 3, 1, 1
     • Eigenvectors: \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}, \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}$$
   • Problem 2:
     • Matrix: $A = \begin{pmatrix} 8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{pmatrix}$
     • Eigenvalues: 0, 3, 15
     • Eigenvectors: \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}, \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$
   • Problem 3:
     • Matrix: $A = \begin{pmatrix} 3 & 10 & 5 \\ -2 & -3 & -4 \\ 3 & 5 & 7 \end{pmatrix}$
     • Eigenvalues: 3, 2, 2
     • Eigenvectors: \begin{pmatrix} 5 \\ 2 \\ -5 \end{pmatrix}$$
2. Multiple-Choice Question (MCQ):
   • Question 1: Find the characteristic equation for the matrix: $\begin{pmatrix} 3 & 10 & 5 \\ -2 & -3 & -4 \\ 3 & 5 & 7 \end{pmatrix}$
   • Answer Options: $\text{(a) } \lambda^3 - 7\lambda^2 - 12\lambda + 16 = 0$ $\text{(b) } \lambda^3 - 7\lambda^2 + 12\lambda - 16 = 0$ $\text{(c) } \lambda^3 - 7\lambda^2 + 16\lambda - 12 = 0$ $\text{(d) } \lambda^3 - 7\lambda^2 - 16\lambda + 12 = 0$

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I can explain how to find the eigenvalues, eigenvectors, or the characteristic equation for these matrices in detail. Let me know if you'd like a deeper dive into any specific part.

Related Questions:

1. How can you verify if a given vector is an eigenvector of a matrix?
2. What is the significance of the determinant when finding eigenvalues?
3. How do you interpret the geometric meaning of eigenvectors and eigenvalues?
4. How can you find the characteristic equation of a 3x3 matrix step-by-step?
5. What are the conditions under which a matrix has repeated eigenvalues?

Tip: When dealing with eigenvalue problems, always check the multiplicity of the eigenvalues as it can affect the number of independent eigenvectors, leading to different solution strategies.