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    $\begingroup$ You can use LinearAlgebra`MatrixConditionNumber to find the condition number of your matrix. What value you choose as the threshold depends on your particular application... In general, a condition number that is $\mathcal{O}(10^n)$ can make you lose up to $n$ digits of accuracy (in addition to FP errors). $\endgroup$ Commented Feb 16, 2013 at 19:28
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    $\begingroup$ Numerical inversion of a matrix is both slow and poorly conditioned, a fact mentioned in most books on numerical analysis. Typically, a solution to an applied problem can be formulated in such a way that matrix inversion is expressed in terms of the solution of a system of equations. Naturally, it would be much easier to provide details if you provided more details of your problem. $\endgroup$ Commented Feb 16, 2013 at 19:36
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    $\begingroup$ If you already happen to know the eigenvalues or singular values of the matrix, then compare the one of smallest absolute value to the one of largest absolute value: that ratio is the reciprocal of the condition number. Wikipedia gives some other alternative calculations for special circumstances. $\endgroup$ Commented Feb 16, 2013 at 19:44