2. (20 points) Consider a discrete-time linear dynamical system with state $x \in \mathbb{R}^3$, described by the linear equations
$x(t+1) = Ax(t)$, $t = 0, 1, 2, ...$
$\begin{pmatrix} -3 & 1 & 2 \\ 0 & 2 & 5 \\ 0 & 0 & -2 \end{pmatrix}$
with $A = \begin{pmatrix} -3 & 1 & 2\\ 0 & 2 & 5\\ 0 & 0 & -2 \end{pmatrix}$.
(a) (10 points) Find all the eigenvalues of A and determine whether the system is asymptotically stable.
(b) (10 points) We add a control unit $u(t) \in \mathbb{R}^3$ such that
$x(t+1) = Ax(t) + u(t)$, $t = 0, 1, 2, ...$
$\begin{pmatrix} m_1 & 2 & 3 \\ 0 & m_2 & 1 \\ 0 & 0 & m_3 \end{pmatrix}$If $u(t) = \begin{pmatrix} m_1 & 2 & 3\\ 0 & m_2 & 1\\ 0 & 0 & m_3 \end{pmatrix} x(t)$, what are the conditions for $m_1, m_2, m_3$ that make the system asymptotically stable?
