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

Here’s a summary of the problem:

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.

Solution overview:

(a) To find:

• det(A): The determinant of a matrix is the product of its eigenvalues. $\det(A) = \lambda_1 \cdot \lambda_2 \cdot \lambda_3 = 1 \cdot 1 \cdot 0.5 = 0.5$

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