(Solution) - Exercises 1 and 2 apply to a 3 x 3 -(2025 Original AI-Free Solution)

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Exercises 1 and 2 apply to a 3 x 3 matrix A whose eigenvalues are estimated to be 4, -4, and 3.
1. If the eigenvalues close to 4 and -4 are known to have different absolute values, will the power method work? Is it likely to be useful?
2. Suppose the eigenvalues close to 4 and -4 are blown to have exactly the same absolute value. Describe how one might obtain a sequence that estimates the eigenvalue close to 4.