Please help with questions 2 and 3. ### Linear Algebra: Eigenvalues and Diagonalization#### Problem 2The matrix \( A \) has eigenvalues \(-2\), \(-2\), and \(1\). Find 3 linearly independent eigenvectors for \( A \). Show enough work to demonstrate your understanding of how to solve this problem by hand.The matrix \( A \) is given by:\[ A = \begin{bmatrix}1 & 3 & 3 \\-3 & -5 & -3 \\3 & 3 & 1\end{bmatrix}\]#### Solution Approach:1. **Step 1: Eigenvalue Calculation:** - The eigenvalues are provided: \(-2\), \(-2\), and \(1\).2. **Step 2: Eigenvector Calculation for \( \lambda = -2 \):** - Solve \((A - (-2)I) \mathbf{v} = 0\) to find the eigenvectors associated with \( \lambda = -2 \).3. **Step 3: Eigenvector Calculation for \( \lambda = 1 \):** - Solve \((A - I) \mathbf{v} = 0\) to find the eigenvectors associated with \( \lambda = 1 \).#### Problem 3Diagonalize the matrix in part 1a. Again, show enough work to demonstrate your understanding of how to solve this problem by hand.#### Solution Approach:1. **Step 1: Matrix of Eigenvectors \( P \):** - Construct a matrix \( P \) where each column is an eigenvector of \( A \).2. **Step 2: Diagonal Matrix \( D \):** - Construct the diagonal matrix \( D \), where the diagonal elements are the eigenvalues of \( A \).3. **Step 3: Verification:** - Verify that \( A = PDP^{-1} \).Ensure to demonstrate each step clearly and provide calculations to show your method and understanding.

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Answered: 2. The matrix A has eigenvalues -2,-2,…

2. The matrix A has eigenvalues -2,-2, and 1. Find 3 linearly independent eigenvec- tors for A. Show enough work to show you know how to do this problem by hand. 3 3

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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