Consider a SISO system state equation:
$$
\mathbf{\dot{x}} = \underbrace{\begin{bmatrix}
\eta & 1 & 0 & \dots & 0 \\
0 & \eta & 1 & \dots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \dots & 1 \\
0 & 0 & 0 & \dots & \eta
\end{bmatrix}}_{\mathbf{A}} \mathbf{x} + \underbrace{\begin{bmatrix}
b_1 \\
b_2 \\
\vdots \\
b_{n-1} \\
b_n
\end{bmatrix}}_{\mathbf{B}} u
$$
$$
y = \underbrace{\begin{bmatrix}
c_1 & c_2 & \dots & c_{n-1} & c_n
\end{bmatrix}}_{\mathbf{C}} \mathbf{x}
$$
(a) Show that (A, B) is controllable if and only if $b_n \ne 0$. (5%)
(b) Show that (A, C) is observable if and only if $c_1 \ne 0$. (5%)