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Nice write up! I'd caution just checking for an exactly-zero determinant. Read up about ill-conditioned matrices, and maybe check that conditioning number (or a determinant below a certain threshold) first. Also, work hard to never, ever have to actually fully invert a matrix.

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Checking that the determinant is below a threshold is not a valid test for conditioning. Take epsilon*Identity, for example.

As lp251 says, the determinant method is not a good idea.

Checking the condition number of the covariance matrix of the LS estimates (X'X)^{-1} is the way R does it.

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