I have a system o linear equations dependent on parameters $a,b \in \mathbb{C}$. I am not quite sure what to do with it. My only intuition is, that if I were to use Gaussian elimination to get the pivots (from column space), I could somehow determine what are the conditions of $a$ and $b$. Is it on the right track, or can someone help me find a better way of looking at it?
$\begin{bmatrix} -i & a & 1+2i &\bigm| & 0 \\ 1 & 3i & b & \bigm| & 0 \\ i & -3 & -i & \bigm| & -1 \end{bmatrix}$
Note: The word "parameter" is used here in a different sense than "free variable". By assignment it is meant that for each choice of complex numbers $ a, b $ you are to find all solutions of a given system of three equations with three unknowns. It is possible that the existence, number of solutions, etc., depends on $ a, b $ - let's discuss. We need to solve the system for all choices of $ a, b $.