Determine system controllability based on solutions to the state equation with zero input Given is a single input single output, time invariant state space system.
\begin{equation}
x(t) = \left(\begin{array}{r} 5 \\ -1 \\ 4\end{array}\right)e^{-2t} 
\end{equation}
\begin{equation}
x(t) = \left(\begin{array}{c} 3 \\ 3 \\ 1 \end{array}\right)e^{-2t} 
\end{equation}
\begin{equation}
x(t) = \left(\begin{array}{r} 6 \\ 2 \\ -7\end{array}\right) e^{-t} 
\end{equation}
The above state trajectories arise from different initial states at time $0$. In each case the input to the system $u(t)=0$. 
How do you determine whether the system is controllable or not?
There is not enough information to try to calculate matrix $A$ and $B$ is not given, so the controllability matrix can't be constructed. I am thinking the answer has to do with the fact that the state trajectories do not have the same form (i.e $e^{-2t}$ and $e^{-t}$).
Any ideas?   

You can construct the $A$ matrix because eigenvectors and eigenvalues are given. Like
$$T = \begin{bmatrix} 5 & 3 & 6 \\ -1 & 3 & 2 \\ 4 & 1 & -7 \end{bmatrix}$$
$$\Lambda = \begin{bmatrix} -2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -1 \end{bmatrix}$$
Then, $A = T \Lambda T^{-1}$. Now you can show that the system is not controllable for any $b^T = \begin{bmatrix} b_1 & b_2 & b_3 \end{bmatrix}$.
Note that any $n$th order system is always controllable if $B$ is a rank $n$ matrix so I believe this system has 1 input.
