QuestionThere are ten problems. Solve all problems. You may use the eigenvalues and the linearly independent eigenvectors which you found in Homework 8 to solve the problems in this assignment. However, you must show all supporting work (except for the work to find the eigenvalues and the eigenvectors).
Let $\alpha \neq 0$ and $\beta \neq 0$ be real nonzero constants. Define the following real symmetric matrices (3) $$A_{3}=\left(\begin{array}{cccc} \alpha & -\beta & -\beta & -\beta \\ -\beta & \alpha & -\beta & -\beta \\ -\beta & -\beta & \alpha & -\beta \\ -\beta & -\beta & -\beta & \alpha \end{array}\right)$$
The computations show that there are two cipenyalues $$\lambda_{1}=\alpha-\beta \rho_{2} \quad \lambda_{2}=\alpha+\beta$$
Moreover, the bases of the eigenspaces are given by, respectively $$\begin{array}{l} \mathcal{B}_{1}=\left\{\left(\begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \end{array}\right)\right\} \text { for the eigenvalue } \lambda_{1}=\alpha-3 \beta, \\ \mathcal{B}_{2}=\left\{\left(\begin{array}{c} 1 \\ 1 \\ -1 \\ -1 \end{array}\right),\left(\begin{array}{c} 1 \\ -1 \\ -1 \\ 1 \end{array}\right),\left(\begin{array}{c} 1 \\ -1 \\ 1 \\ -1 \end{array}\right)\right\} \text {, for the eigenvalue } \lambda_{2}-a+B \end{array}$$
Solve the following first order homogeneous systems of differential equations (3) $\frac{\mathrm{d}}{\mathrm{d} t} \mathbf{u}=A_{3} \mathbf{u}$