# Ill-conditioned matrix [duplicate]

Possible Duplicate:
Inverse matrices are close iff matrices are close

Consider this problem:

$Ax = b$

I want to solve it/find x and the matrix A is ill-conditioned. Why is the fact "A is ill-conditioned" a "bad" thing?

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## marked as duplicate by Did, Listing, Ｊ. Ｍ., Ilya, t.b.Oct 28 '11 at 14:38

Because if one knows the coefficients of $A$ only up to a given precision, a small variation in them will cause a huge variation in the coefficients of $A^{-1}$, hence, presumably, in the solution $x=A^{-1}b$. Alternatively, even if $A^{-1}$ is known with an absolute precision, if one knows the coefficients of $b$ only up to a given precision, a small variation in them will cause a notable variation in the coefficients of the solution $x=A^{-1}b$ since some coefficients of $A^{-1}$ are large. In real life, both effects are often conspiring. See the definition of the condition number of a matrix.