(Solved): Problem 4. a) Write down the characteristic polynomial of the matrix \[ A=\left(\begin{array}{ccc} ...

Problem 4. a) Write down the characteristic polynomial of the matrix \[ A=\left(\begin{array}{ccc} 2 & 0 & \frac{1}{2} \\ 0 & 3 & 0 \\ -2 & 0 & 2 \end{array}\right) \] b) Compute the eigenvalues \( \lambda_{1}, \lambda_{2} \) and \( \lambda_{3} \) of the matrix \( A=\left(\begin{array}{ccc}2 & 0 & \frac{1}{2} \\ 0 & 3 & 0 \\ -2 & 0 & 2\end{array}\right) \). c) For each of the eigenvalues \( \lambda_{i}, i=1,2 \) and 3, write down the corresponding eigenvectors equations \[ \left(\begin{array}{ccc} 2 & 0 & \frac{1}{2} \\ 0 & 3 & 0 \\ -2 & 0 & 2 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\lambda_{i}\left(\begin{array}{l} x \\ y \\ z \end{array}\right) \] d) Compute the eigenvectors. e) Assemble the general (possibly complex-number) solution of the linear system \[ \frac{d}{d t}\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\left(\begin{array}{ccc} 2 & 0 & \frac{1}{2} \\ 0 & 3 & 0 \\ -2 & 0 & 2 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right) \] of differential equations.