Determine the controllability and observability for each of the following systems.

a. \(\left\{\begin{array}{l}\dot{x}_{1} \\ \dot{x}_{2} \\ \dot{x}_{3}\end{array}\right\}=\left[\begin{array}{ccc}-5 & -3 & 0 \\ 2 & 0 & 0 \\ 0 & 1 & 0\end{array}\right]\left\{\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right\}+\left[\begin{array}{l}2 \\ 0 \\ 0\end{array}\right] u, \quad y=\left[\begin{array}{lll}0 & 1 & 6\end{array}\right]\left\{\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right\}\)

b. \(\left\{\begin{array}{l}\dot{x}_{1} \\ \dot{x}_{2} \\ \dot{x}_{3}\end{array}\right\}=\left[\begin{array}{ccc}-1 & -1 & -2 \\ 4 & 0 & 0 \\ 0 & 0 & 2\end{array}\right]\left\{\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right\}+\left[\begin{array}{l}2 \\ 0 \\ 0\end{array}\right] u, \quad y=\left[\begin{array}{lll}2 & 2 & 1\end{array}\right]\left\{\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right\}\)

c. \(\left\{\begin{array}{l}\dot{x}_{1} \\ \dot{x}_{2} \\ \dot{x}_{3}\end{array}\right\}=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 7 & -6\end{array}\right]\left\{\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right\}+\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right] u, \quad y=\left[\begin{array}{lll}1 & 1 & 0\end{array}\right]\left\{\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right\}\)

d. \(\left\{\begin{array}{l}\dot{x}_{1} \\ \dot{x}_{2} \\ \dot{x}_{3}\end{array}\right\}=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 4 & 3\end{array}\right]\left\{\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right\}+\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right] u, \quad y=\left[\begin{array}{lll}1 & 1 & 1\end{array}\right]\left\{\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right\}\)