Solving a system of matrices

Hello,

I am trying to solve a set of equations in matrix form for a multiple input and output network.

Here is how I declare it:

    S = [1 1 1 1; 1 1 1 1; 1 1 1 1; 1 1 1 1];

    syms A B C D A4 C1;

    A = [1; 0; 0; A4];
    B = S*A;
    C = [C1; 0; 0; 0];
    D = S*C;
    A4 = 0.5*D(1);
    C1 = 0.5*B(4);

I simplified all constants to simple 1's and 0's for sake of example. My real purpose is of course more complex.

If the matrices are expanded, we get a total of 10 unique equations for 10 unknowns. I have solved this by hand and could technically solve it through the 'solve' command if I write out all the equations.

I would like to know if there is a more elegant method of solving it, as the number of variables becomes ridiculous for bigger networks.

Thanks!