I need to find a relation between $a$, $b$, $c$, $d$ in order the system with the following augmented matrix has at least one non-trivial solution. I have tried both the Gaussian and Gauss-Jordan elimination methods, but I think it is gets too complicated in the end. Thanks in advance.
$$ \left[ \begin{array}{c|c} A&b\\ \end{array} \right] = \left[ \begin{array}{cccc|c} 2&3&-1&4&a\\ 3&1&-1&0&b\\ 0&2&1&1&c\\ -1&0&-1&3&d \end{array} \right] $$
On
Don't forget that you can swap rows if it's convenient, not only if it's necessary. Doing this, the row-reduction becomes very simple: $$\pmatrix{-1&0&-1&3&|&d\cr 0&2&1&1&|&c\cr 0&3&-3&10&|&a+2d\cr 0&1&-4&9&|&b+3d\cr} \sim\pmatrix{-1&0&-1&3&|&d\cr 0&1&-4&9&|&b+3d\cr 0&0&9&-17&|&c-2b-6d\cr 0&0&9&-17&|&a-3b-7d}$$ and I'm sure you can finish it from here.
Gaussian elimination produces $$ \left[ \matrix{1 & 3/2 & -1/2 & 2 & | & a/2\cr 0 & 1 & -1/7 & 12/7 & | & 3a/7 -2b/7\cr 0 & 0 & 1 & -17/9 & | & -2a/3 + 4b/9 +7c/9 \cr 0 & 0 & 0 & 0 & | & -a+b+c+d\cr}\right]$$
so the condition for a solution is $-a+b+c+d=0$.