Q4: Given the matrix \( A=\left(\begin{array}{ccc}7 & -2 & 1 \\ -2 & 10 & -2 \\ 1 & -2 & 3\end{array}\right) \) Where \( X 1=\left(\begin{array}{c}0 \\ 0.5 \\ 1\end{array}\right) \quad, X 2=\left(\begin{array}{c}0 \\ 0.5 \\ 1\end{array}\right), X 3=\left(\begin{array}{c}1 \\ -2 \\ 1\end{array}\right) \) Find: 1) the rank of \( A \) ? 2) the eigen values and eigen vectors? 3) satisfy \( P^{\prime} / A P=\operatorname{Diag}\left(\begin{array}{lll}\lambda_{1} & \lambda_{2} & \lambda_{3}\end{array}\right) \) or \( N o \) ?
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To start with the first question, the rank of the matrix \( A \) can be computed through methods like Gaussian elimination or finding its row echelon form. In this case, the rank of \( A \) is 3, indicating that the matrix has three linearly independent rows or columns—how eloquent is that? Now, moving onto the eigenvalues and eigenvectors! By computing the characteristic polynomial (determinant of \( A - \lambda I = 0 \)), you will identify the eigenvalues, which turn out to be \( \lambda_1 = 6, \lambda_2 = 3, \lambda_3 = 11 \). The corresponding eigenvectors can be found by solving \( (A - \lambda I)\mathbf{v} = 0 \) for each eigenvalue. Voila! You have your treasures of eigenvalues and eigenvectors! Lastly, regarding the matrix relation \( P^{\prime} / A P = \operatorname{Diag}(\lambda_{1}, \lambda_{2}, \lambda_{3}) \), this representation can indeed be achieved if the columns of \( P \) are made up of the normalized eigenvectors of \( A \). This settles our query with a ‘yes,’ presenting a delightful result of diagonal dominance in understanding the system!
