the first equation does NOT imply a = b = 0, it implies a+b = 0, that is b = -a.
the second equation implies b = a, which TOGETHER with the first, implies a = -a, so 2a = 0, so a = 0/2 = 0.
since b = a, b = 0 as well.
your edit is perfectly good reasoning. but we can solve for r,s explicitly, as well:
ri + si = a --> r + s = -ai. adding that to the second equation gives:
2r = b-ai --> r = (b-ai)/2.
then (b-ai)/2 - s = b, so
s = (b-ai)/2 - b = (b-ai)/2 - 2b/2 = (-b-ai)/2