I need to freshen up on my linear algebra. I don’t remember on which sense it goes but I think if you swap the factors, you’ll have to you transpose the matrix x to keep the same result.

You are right. I will correct it.

That matrix of zeros with one in diagonal is called the matrix of identity.
It is famous because when doing multiplication on matrix or vector, it acts likes 1 on “normal” number:
x times 1 is x anyMatrix times Identity is anyMatrix.

Thank you OP!