[ChrisFoster]

My main concern with this comparison is that there's no mention of numerical stability / accuracy of this method.

Calculating determinants is notoriously prone to cancellation error if done naively, so you've got to be really careful. By extension, the usual text book implementation of Cramer's rule is a rather bad way to solve matrix systems. (I don't know much about Grassman.jl so perhaps they've used some known tricks to make this work better, but you do have to be really careful with this stuff.)

Actually in early versions of StaticArrays we were pretty cavalier about numerical accuracy because it was just a library we needed for fast geometric calculations. However as people started using this library for serious numerical work we've had to pay a lot more attention to having good numerical properties.

There's still a lot to learn - and no doubt much work to do - in upgrading the implementation to make sure everything is first robust, and second as fast as possible.

[DreamScatter]

This algorithm is not Cramer's rule, it is analogous to it, but based on the original exterior algebra of Hermann Grassmann.

[ChrisFoster]

I agree the exterior algebra is beautifully elegant mathematics, but expressive elegance of the formalism is largely irrelevant to numerical robustness. (Cramer's rule itself is a perfect example of this: it's elementary to express, but in simple form is a terrible idea numerically.)

To be clear, I'm not claiming there's anything wrong with your algorithm but simply adding a note of caution. Any careful comparison between methods must include both efficiency and numerical accuracy.