Couldn't find a similar problem in previous questions. I'm working on a control systems problem in state space. We are given the $A$ and $B$ matrices as well as our inputs and states. To determine BIBO stability, we need to determine our $C$ matrix, which we assume is $[c_1\;c_2\;c_3]$, and we assume our $D$ matrix is $[0]$. From earlier in the problem, we have determined that our system is marginally stable by calculating the eigenvalues of $A$ using $\text{det}(\lambda I-A)$. I'm currently stuck on finding the coefficients $c_1,c_2,c_3$, and not sure if there is something trivial I might be missing.
For reference, the $A$ matrix is $\begin{bmatrix}0 & 1 & 0\\ 0& 0& -g\\ 0& 1/R & 0\end{bmatrix}$, where $g$ is the acceleration due to gravity and $R$ is the radius of the earth. The $B$ matrix is $\begin{bmatrix}0 & 0\\ 1 & 0\\ 0 & 1\end{bmatrix}$.
Any help would be greatly appreciated.
To give you a clue about what is going on: The standard state space model $(A,B,C,D)$ has \begin{align} G(s) = \frac{Y(s)}{U(s)}=C(sI-A)^{-1}B+D \end{align} as its transfer function. The $C$ matrix does not influence stability of the entire system, $A$ does. You can however select certain stable/unstable modes, but by doing so, you could trick yourself. For example, take the system \begin{align}&\frac{d}{dt} \begin{pmatrix} x_1\\ x_2 \end{pmatrix}= \begin{pmatrix} A_{11} & 0\\ 0 & A_{22} \end{pmatrix}\begin{pmatrix} x_1\\ x_2 \end{pmatrix} + \begin{pmatrix} B_1\\ 0 \end{pmatrix}u\\ &y = \begin{pmatrix} C_1 & 0 \end{pmatrix}\begin{pmatrix} x_1\\ x_2 \end{pmatrix}. \end{align} Now assume $A_{11}$ is stable and $A_{22}$ is unstable. Have a look at the transfer function \begin{align} G(s) &= \begin{pmatrix} C_1 & 0 \end{pmatrix} \begin{pmatrix} (sI-A_{11})^{-1} & 0\\ 0 & (sI-A_{22})^{-1} \end{pmatrix} \begin{pmatrix} B_1\\ 0 \end{pmatrix}\\ &=C_1(sI-A_{11})^{-1}B_1. \end{align} So in the end, the input-output map is BIBO stable, but internally things might blow up(state $x_2$).