GATE EC 2012 Set 1 Q32 — LTI Systems

The state variable description of an LTI system is given by

$$\begin{pmatrix} \dot{x_1}\\ \dot{x_2}\\ \dot{x_3} \end{pmatrix}=\begin{pmatrix} 0 & a_1 & 0\\ 0 & 0 & a_2\\a_3 & 0 & 0 \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\x_3 \end{pmatrix}+\begin{pmatrix} 0\\ 0\\1 \end{pmatrix}u$$

$$y=\begin{pmatrix} 1 &0&0\end{pmatrix}\begin{pmatrix} x_1\\ x_2\\x_3 \end{pmatrix}$$

where $y$ is the output and $u$ is the input. The system is controllable for