$1-$ Let $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 1\\ 0& 1& 2\end{array}\right]$

$($a$)$ calculate $A^{100}$

$($b$)$ calculate $e^{At}$

$2-$ Consider the following system:

Where the damping constant $b=4,$ the spring coefficient $k=2,$ and the mass $M=1kg.$

a$)$ Find the state space of the system.

b$)$ Find the transfer function of the system directly from the system's differential equations.

c$)$ Find the transfer function of the system from the state$-$space equations achieved in $($a$).$