Word Count : 712

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To diagonalize the matrix \( A = \begin{bmatrix} 2 & 0 & 0 \\ 1 & 2 & -1 \\ 1 & 3 & -2 \end{bmatrix} \), we need to find the eigenvectors corresponding to each eigenvalue, then use these eigenvectors to form the matrix \( P \). The diagonal matrix \( D \) will consist of the eigenvalues of \( A \). ### Steps: 1. Eigenvalues: Given as \( \lambda_1 = 2 \), \( \lambda_2 = 1 \), and \( \lambda_3 = -1 \). 2. Eigenvectors: We solve \( (A - \lambda I)v = 0 \) for each eigenvalue \( \lambda \), where \( I \) is the identity matrix and \( v \) is the eigenvector corresponding to \( \lambda \). Let's start with each eigenvalue and compute the corresponding eigenvector. ### For \( \lambda_1 = 2 \): Solve \( (A - 2I)v = 0 \): \[ A - 2I = \begin{bmatrix} 2 & 0 & 0 \\ 1 & 2 & -1 \\ 1 & 3 & -2 \end{bmatrix} - \begin{bmatrix} 2 & 0 & _____ ________ ______ ________ _______ ______ ______ __________ ________ _______ ___ _____.
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