I have been taught that "inverting a square matrix with small determinant is numerically unstable because it is close to singular"?
Is this right opinion?
2026-04-07 20:03:41.1775592221
Relation of numerical stability of matrix inversion and it's determinant
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In some sense. It's more so that you matrix cannot be both large and small. If you think about it, having a small determinant means the inverse will be divided by a small number, so you effectively are multiplying a bunch of calculations and their error by a huge number, so the error becomes large.
A way to estimate the error is from the condition number. There's a lot of formule on that page, but you can think about just using the largest singular value over the smallest singular value, or the largest eigenvalue over the smallest eigenvalue. That tells you the relative scales that values in your matrix cover, and if they cover a large scale, then things can get ugly.