The following system is given and I'm asked to find the transfer function $$\frac{Y(s)}{U(s)}=G(s)$$ $$\bar {\dot x}=\begin{bmatrix} 0 & 1 & 0 & 0 &0 \\0 & 0 & 1 & 0 & 0 \\ 0 & -1 & -2 & 0 & 0 \\ 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & -3\end{bmatrix} \bar x + \begin{bmatrix} 0\\0\\1\\1\\0\end{bmatrix}u=A\bar x + Bu \\ $$

$$y=\begin{bmatrix}0 & 1 & 0 & 0 & 1\end{bmatrix} \bar x $$

I haven't practised that much on state space models and I don't remember that much from matrix algebra but here's what I've thought.

I found the eigenvalues seeing the matrix as a 2x2 diagonal matrix with matrix elements therefore getting the denominator of my transfer function $$(s+1)^2(s+2)(s+3)$$.

I also have my backdoor $$y=C(sI-A)^{-1}B $$ but this includes many calculations and I guess that there is a simpler solution.