Also, it seems the author solved a triangular system. LAPACK has special routines for that. Were they used?

LAPACK is typically optimized for larger systems, not 5 unknowns. Also, 5 is not a great number for vectorized operations - it might even be beneficial to zero/one pad the matrix.

Optimized LAPACK is often 5-10x faster than «basic LAPACK».

BLAS/LAPACK was originally written in the 70s/80s and sometimes unroll loops to the tune of 5 or 7 - it made sense then. Not so much these days.

I didn’t read the article in detail, but there appear to be a lot of holes in it.

Personally, my biggest reason to prefer LAPACK in general is that its authors have already put a great deal of effort into correctness and numerical stability, so I don't have to. Even basic LAPACK is pretty fast, let alone the optimized libraries. Hand-optimizing my own special case is an absolute last resort.

Yes, Cramer’s rule (referred to in the article) does not give numerically stable results AFAIK - which is yet another can of worms. Good catch.

I don't think I have heard Cramer's rule mentioned without the caveat "but don't use this for real problems".

It’s useful for certain theoretical proofs in analytic math - or so I’ve heard. Not all math is well suited to be coded into reliable software.