Let's say I have the following equation $Ax=b$
My question is - how can i find a vector $b$ around which the above equation is not stable?
I have $A= \begin{bmatrix} 1 & 0.999 \\ 1 & 1.001 \\ \end{bmatrix} $
This is what i have done so far. $x=A^{-1}b$ and $A^{-1}= \begin{bmatrix} 500.5 & -499.5 \\ -500 & 500 \\ \end{bmatrix} $ So I see that the $\|A^{-1}\|=1000$ is very large. I am guessing that $\|b\|$ should be very small, but I do not know how to EXACTLY FORMALLY formulate the solution to this problem.
Thanks