QuestionLet $\alpha \neq 0$ and $\beta \neq 0$ be real nonzero constants. Define the following real symmetric matrices (1) $A_{1}=\left(\begin{array}{ccc}\alpha & -\alpha & \alpha \\ -\alpha & \alpha & \alpha \\ \alpha & \alpha & \alpha\end{array}\right)$, (2) $A_{2}=\left(\begin{array}{ccc}\alpha & -\beta & -\beta \\ -\beta & \alpha & -\beta \\ -\beta & -\beta & \alpha\end{array}\right)$, (3) $A_{3}=\left(\begin{array}{cccc}\alpha & -\beta & -\beta & -\beta \\ -\beta & \alpha & -\beta & -\beta \\ -\beta & -\beta & \alpha & -\beta \\ -\beta & -\beta & -\beta & \alpha\end{array}\right)$, (4) $A_{4}=\left(\begin{array}{cccc}\alpha & -\beta & \alpha & -\beta \\ -\beta & \alpha & -\beta & \alpha \\ \alpha & -\beta & \alpha & -\beta \\ -\beta & \alpha & -\beta & \alpha\end{array}\right)$ (5) $\quad A_{5}=\left(\begin{array}{lllll}\alpha & \beta & 0 & 0 & 0 \\ 0 & \alpha & \beta & 0 & 0 \\ 0 & 0 & \alpha & \beta & 0 \\ 0 & 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & 0 & \alpha\end{array}\right)$
Solve the following first order homogeneous systems of differential equations (1) $\frac{\mathrm{d}}{\mathrm{d} t} \mathrm{u}=A_{1} \mathbf{u}$ (2) $\frac{\mathrm{d}}{\mathrm{d} t} \mathbf{u}=A_{2} \mathbf{u}$ (3) $\frac{\mathrm{d}}{\mathrm{d} t} \mathrm{u}=A_{3} \mathrm{u}$ (4) $\frac{\mathrm{d}}{\mathrm{d} t} \mathbf{u}=A_{4} \mathbf{u}$ (5) For the system $\mathbf{u}^{\prime}=A_{5} \mathbf{u}$, find a solution of the form $$\mathbf{u}(t)=e^{a t}(\xi+\eta t)$$ where $\xi$ and $\eta$ are real nonzero constant vectors. Let $\alpha \neq 0, \beta \neq 0$ and $\gamma \neq 0$ be real nonzero constants. Let $\mu$ be a real constant. Define the following real matrices (1) $B_{1}=\left(\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right)$, (2) $B_{2}=\left(\begin{array}{cc}0 & \beta \\ -\beta & 0\end{array}\right)$, (3) $B_{3}=\left(\begin{array}{cc}\alpha & \beta \\ -\beta & \alpha\end{array}\right)$, (4) $B_{4}=\left(\begin{array}{ccc}0 & \alpha & \beta \\ -\alpha & 0 & \gamma \\ -\beta & -\gamma & 0\end{array}\right)$ (5) $B_{5}=\left(\begin{array}{ccc}\mu & \alpha & \beta \\ -\alpha & \mu & \gamma \\ -\beta & -\gamma & \mu\end{array}\right)$.
Solve the first order homogeneous linear differential equations (6) $\frac{\mathrm{d}}{\mathrm{d} t} \mathbf{u}=B_{1} \mathbf{u}$ (7) $\frac{\mathrm{d}}{\mathrm{d} t} \mathbf{u}=B_{2} \mathbf{u}$ (8) $\frac{\mathrm{d}}{\mathrm{d} t} \mathrm{u}=B_{3} \mathrm{u}$ (9) $\frac{\mathrm{d}}{\mathrm{d} t} \mathbf{u}=B_{4} \mathbf{u}$ (10) $\frac{\mathrm{d}}{\mathrm{d} t} \mathbf{u}=B_{5} \mathbf{u}$