$$M = \left(\begin{smallmatrix} a_1 & a_2 & a_3 & a_4\\ b_1 & b_2 & b_3 & b_4\\ a_1 & c_2 & b_2 & c_4\\ a_4 & d_2 & b_3 & c_4\\ b_1 & c_2 & a_2 & e_4\\ b_4 & d_2 & a_3 & e_4\end{smallmatrix}\right)$$ All of the equations equal to 26; augmented, the matrix would then have have "26" to the right of each row.
This is basically the "Star of David" that another user posted:
http://i.stack.imgur.com/mVHwZ.jpg
but I don't think anyone has solved it like this.
Best Answer
You have forgotten the very important condition that all integers between 1 and 12 must be used exactly once.
Though the equations themselves are linear and contain only 0 and 1 as coefficients, the constraint is very tough. I don't think you could find an easy solution without integer programming, which is, like you may have known, NP-Hard and thus we have by now only exponential solutions, which are not better than brute force for such a small problem.