For an invertible square matrix and an induced norm, $\kappa(A)=\|A\|\|A^{-1}\|$. In the 2-norm, $\kappa_2(A)=\sigma_{\max}/\sigma_{\min}\ge1$. Set it to infinity for a singular matrix. A large value measures worst-case sensitivity; it does not mean every input perturbation is amplified maximally.
If only $b$ changes in $Ax=b$, relative solution error is at most $\kappa(A)$ times relative right-side error for nonzero $b$. Conditioning is distinct from algorithmic stability. Scaling all of $A$ by one nonzero scalar leaves its condition number unchanged.
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