I have the following system:
\begin{equation} \begin{split} \dot{x}_{1} &= -\mu x_{1}(t) + b_{11} x_{3}(t) + b_{12} x_{2}(t-\tau_{1}) \\ \dot{x}_{1} &= -\mu x_{2}(t) + b_{21} x_{1}(t-\tau_{1}) + b_{22} x_{4}(t) \\ \dot{x_{3}} &= \mu x_{1}(t-\tau_{1}) - \mu x_{3}(t)\\ \dot{x_{4}} &= \mu x_{2}(t-\tau_{1}) - \mu x_{4}(t)\\ \end{split} \end{equation}
When we try to find its eigenvalues the determinant matrix is written as:
\begin{equation} \begin{split} \begin{vmatrix} \lambda + \mu & -b_{12} e^{\lambda \tau_{1}} & -b_{11} & 0\\ -b_{21} e^{-\lambda \tau_{1}} & \lambda + \mu & 0 & -b_{22} \\ -\mu e^{-\lambda \tau_{1}} & 0 & \lambda + \mu & 0 \\ 0 & -\lambda e^{-\lambda \tau_{1}} & 0 & \lambda + \mu \\ \end{vmatrix} &= 0 \end{split} \end{equation}
I am wondering why exponential map involved here and what is the matrix of the system exactly other than $det(\lambda I -A)$. I couldn't find an explanation for it. It is probably a very basic thing. Yet, I am struggling to understand it.