This code works fast

[glanzkaiser]

This code works fast

[AugustoCL]

Thanks for your words.

But unfortunely, you are misunderstood some important concepts of the Julia Language, like the pre-compilation time because of the Just-In-Time (JIT) compilation paradigm (I really suggest you read about it, it's the main reason that Julia it's so fast).
If you look at your printscreen, 99% of the 5.77 seconds it was consumed by the compilation time because you just use the macro @time, instead of use the @Btime macro from BenchmarkTools.jl package.

Also, since my code has an educational purpose, I didn't dive so deep about the performance of my code. So if you compare correctly the performance, you will see that the \ function has a better performance, and has less alocation (because my function return the full matrix and the \ return just the vector result).