I think degrees of freedom in solving a matrix system work like this: suppose you have an underdetermined matrix (not a contradictory system with two different quantities equal to the same quantity). When you end up solving for x, you get, usually, a vector plus t times another vector. That would be one degree of freedom (t), which is what you get if you have one fewer equations than unknowns. If you have a consistent system with two fewer equations than unknowns, you would get a vector plus t times a vector plus s times another vector. That would be two degrees of freedom. So, to solve your problem, you need to find out which values of a give you redundant lines in your matrix.