Well I am not interested in the solution which is trivial to obtain but in the quality of future solutions.
Let me put it this way:
There are a total of H locations at which observations can be made.
Then if I have initially N locations in A (N >=4) at a given time then which of location from set H that are not in A will be most optimal for addition to A at N+1.
That is when you use condition numbers to test the quality of possible future solutions for addition into A.
So the point is to test the quality of the future solution not using location in A but using the observation in B. By checking which one observation from H can improve the quality of future solution without ever solving the least square itself.
However checking condition number of B would be incorrect as it is not involved in inversion when finding X.
So there must a reason to relate condition number of A and B together for this linear system.