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The image you provided includes Problem 4, a linear algebra question focusing on matrix diagonalization and eigenvalues.

Here’s a summary of the problem:

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**Problem 4:**

Let \( A \) be a \( 3 \times 3 \) matrix with eigenvalues \( \lambda_1 = 1 \), \( \lambda_2 = 1 \), and \( \lambda_3 = 0.5 \), and corresponding eigenvectors \( v_1 = (0, 1, 0) \), \( v_2 = (1, -1, 0) \), and \( v_3 = (1, 1, 1) \). Let \( P_A(\lambda) \) denote the characteristic polynomial of \( A \).

**Questions:**
1. Find \( \det(A) \), \( \text{tr}(A) \), and determine \( P_A(\lambda) \).
2. Determine the geometric multiplicity of each of the eigenvalues of \( A \). Justify your answer.
3. Explain why \( A \) must be diagonalizable, and compute \( A^n \), where \( n \geq 1 \) is an integer.

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**Solution overview:**

(a) To find:
- **det(A)**: The determinant of a matrix is the product of its eigenvalues.
  \[
  \det(A) = \lambda_1 \cdot \lambda_2 \cdot \lambda_3 = 1 \cdot 1 \cdot 0.5 = 0.5
  \]

- **

Let A be a 3x3 matrix with eigenvalues λ1 = 1, λ2 = 1, λ3 = 0.5, and corresponding eigenvectors v1 = (0, 1, 0), v2 = (1, -1, 0), v3 = (1, 1, 1). Compute det(A), tr(A), the characteristic polynomial, geometric multiplicity of eigenvalues, and diagonalize the matrix. Also, compute A^n for n ≥ 1.

Solve a problem involving the diagonalization of a 3x3 matrix with given eigenvalues and eigenvectors. Learn how to compute the determinant, trace, characteristic polynomial, and powers of the matrix A^n for n ≥ 1. Suitable for undergraduate students studying linear algebra.

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