# Equivalent to a condition number inequality but for singular matrices

When we have a linear system ($$AX=b$$) ($$A$$ an invertible matrix of size $$n\times n$$ and $$X,b$$ vectors of size $$n$$) and there's a disturbance in $$b$$ (say $$b+\delta b$$) we get $$A(X+\Delta X)=(b+\delta b)$$ and we have an upper bound for $$\| \Delta X\|/\|X\|$$ using $$K(A)$$, the condition number of $$A$$.

My question is: given the same linear system ($$AX=b$$) but with $$A$$ a singular matrix (so the solutions will take the form of equations), can we find what sort of disturbance will be spread to the solutions, so what will happen to the equations, when disturbing $$b$$, the right-hand term?