Solving Entries in a Four Port Transmission Matrix
Transmission matrices can be used only for two-port networks, since they relate the input on one side to the output on the other and cannot account for more than that. They were conceived to help in the modeling of cascaded two-port networks, since multiplication of transmission matrices results in the transmission matrix of an equivalent two-port network.

What you can do for your four-port network is modeling it with a scattering matrix (https://en.wikipedia.org/wiki/Scattering_parameters) such as:
\begin{equation} 
\begin{pmatrix}
    V_1 \\
    V_2 \\
    V_3 \\
    V_4
\end{pmatrix} = 
\begin{pmatrix}
 S_{11} &amp; S_{12} &amp; S_{13} &amp; S_{14} \\
 S_{21} &amp; S_{22} &amp; S_{23} &amp; S_{24} \\
 S_{31} &amp; S_{32} &amp; S_{33} &amp; S_{34} \\
    S_{41} &amp; S_{42} &amp; S_{43} &amp; S_{44}
\end{pmatrix}
\begin{pmatrix}
    V_1 \\
    V_2 \\
    V_3 \\
    V_4
\end{pmatrix}.
\end{equation}
With this approach, you can simply superpose each input (considering all the others as 0) to determine its contribution to any output (including itself): \$S_{ij}=\frac{V_i}{V_j}\Big|_{V_k=0,\ k\neq j}\$.