Say that we have the system of differential equations in matrix formulation:
$$ \begin{bmatrix} C(t) \\ C_p(t) \end{bmatrix}' = \begin{bmatrix} -k_{cp}-k_{ce} & k_{pc} \\ k_{cp} & - k_{pc} \end{bmatrix} \begin{bmatrix} C(t) \\ C_p(t) \end{bmatrix} + \begin{bmatrix} k_aF \\ 0 \end{bmatrix}e^{-k_at},\;\;\;\; \begin{bmatrix} C(0) \\ C_p(0) \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} $$ I have been given the solution to $C(t)$: $$ C(t)=\frac{Fk_aB}{k_a-\lambda}\left(e^{-\lambda t} - e^{-k_a t}\right) + \frac{Fk_aC}{k_a-\mu}\left(e^{-\mu t} - e^{-k_a t}\right),\;\; B, C \in \mathbb{R}\;\;\;\;\; (1) $$ where $\lambda$ and $\mu$ are the eigenvalues to
$$ A=\begin{bmatrix} -k_{cp}-k_{ce} & k_{pc} \\ k_{cp} & - k_{pc} \end{bmatrix}. $$
What I want to do is to somehow determine $k_{cp}$, $k_{pc}$, $k_{ce}$ if I am given equation 1. I know that $$ \lambda+\mu = \text{Tr} A =-k_{cp}-k_{pc}-k_{ce},\\ \mu\lambda = \text{det} A=k_{ce}k_{pc}. $$ My problem is that I need a third equation to be able to solve for $k_{cp}$, $k_{pc}$, $k_{ce}$, but I am not really sure how I can find one more equation. Thanks!
Edit: to be extra clear, say that I have a specific solution and I know all the values of the constans in equation (1). I am seeking a system of equations that out of which I can determine $k_{cp}$, $k_{pc}$, $k_{ce}$ given the solution.