Transmission matrix of a two-port
The trick can be to notice which Kirchhoffs-law to use. I suggest you to try with Kirchhoff's Loop Rule instead of Kirchhoffs current law.

There are two simple loops in which if you calculate each voltages, you will get two equations from which the transmission characteristic (and matrix) can be determined.







simulate this circuit &ndash; Schematic created using CircuitLab

The transmission characteristic looks like:

$$ V_1 = A \times V_2 + B \times I_2  $$
$$ I_1 = C \times V_2 + D \times I_2  $$

The two equations for the two loops:

$$ V_1 = j\omega L \times I_1 + \frac{1}{j\omega C} \times (I_1 + I_2) $$
$$ V_2 = \frac{1}{j\omega C} \times (I_1 + I_2) $$

Your task is to order them to get A,B,C,D. Though, sometimes it can be a little difficult/messy but not in this case.



Furthermore, there is an other method to calculate the transmission matrix of a two-port. There two special case; when \$ V_2 = 0 \$ and when \$  I_2 = 0 \$ , and in these cases siplified equations can be used to determine A,B,C,D:


These two documents describe this method:


First, only about transmission characteristic. 
Second, generally about two-ports, transmission characteristic starts on page 19.


So all in all, you should try to answer the question using one of the methods and the check your results with the other. This way you can practice both of them while making sure that your homework will be correct.