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.