I have matrix $\mathbf{A}=\begin{bmatrix}2 & 1 & 0 & 0 \\ 0 & 2 & 0 &0 \\ 0 & 0 & -1 &0 \\0 & 0 & 0 &-2\end{bmatrix}$ and vector $\mathbf{b}=\begin{bmatrix}0\\1\\1\\2\end{bmatrix}$.

An LTI system is described by the equation $\dot{\mathbf{x}}=\mathbf{Ax}+\mathbf{b}u$ where $u$ is the input of the system.

I want to decide if the sytem is controllable or not without calculating the controllability matrix $\mathcal{M}\_c=\begin{bmatrix}\mathbf{B}&\mathbf{AB}&\mathbf{A^2B}&\mathbf{A^3B}\end{bmatrix}$.

Is there any way to do that?
Thanks