The state variable description of an $\text{LTI}$ system is given by $$\begin{bmatrix}\dot{x_1}\\\dot{x_2\\\dot{x_3}}\end{bmatrix}=\begin{bmatrix}0 & a_1 & 0\\0 & 0 & a_2\\a_3 & 0 & 0\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}+\begin{bmatrix}0\\0\\1\end{bmatrix}u$$
$$y=\begin{pmatrix}1 & 0 & 0\end{pmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}$$
where $y$ is the output and $u$ is the input. The system is controllable for
- $a_1\neq 0,$ $a_2=0,$ $a_3\neq 0$
- $a_1= 0,$ $a_2\neq 0,$ $a_3\neq 0$
- $a_1=0,$ $a_2\neq 0,$ $a_3=0$
- $a_1\neq 0,$ $a_2\neq 0,$ $a_3=0$