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To find the eigenvalues and eigenvectors of the given matrix: \[ A = \begin{bmatrix}-2 & 5 & 4 \\ 5 & 7 & 5 \\ 4 & 5 & -2 \end{bmatrix} \] we follow these steps: ### Step 1: Compute the Characteristic Equation The eigenvalues \(\lambda\) are obtained by solving: \[ \det(A - \lambda I) = 0 \] where \(I\) is the identity matrix. Subtracting \(\lambda I\) from \(A\): \[ A - \lambda I = \begin{bmatrix}-2 - \lambda & 5 & 4 \\ 5 & 7 - \lambda & 5 \\ 4 & 5 & -2 - \lambda \end{bmatrix} \] The determinant is given by: \[ \begin{vmatrix} -2-\lambda & 5 & 4 \\ 5 & 7-\lambda & 5 \\ 4 & 5 & -2-\lambda \end{vmatrix} = 0 \] Expanding along the first row: \[ (-2-\lambda) \begin{vmatrix} 7-\lambda & 5 \\ 5 & -2-\lambda \end{vmatrix} - 5 \begin{vmatrix} 5 & 5 \\ 4 & -2-\lambda \end{vmatrix} + 4 \begin{vmatrix} 5 & 7-\lambda \\ 4 & 5 \end{vmatrix} \] Computing the 2×2 determinants: \[ \begin{vmatrix} 7-\lambda & 5 \\ ________ __________ __________ _________ ________ _____ _____ ___.
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