Does the order in tensor definition count?

Question

I see different books use different order in tensor definition. I can't understand how this happens because after all in the definition of a rank 2 tensor we have matrices multiplications, that in general are not commutative. One can think that this is just a convention, that it is sufficient to keep the same convention everywhere, but probably this is not the case because I see that jacobian matrix are moved freely, for example here: I'm confused. The terms exchanged have common indexes, and in general $a_{ir}b_{rj} \neq b_{ir} a_{rj}$ (or, other words, the $(i,j)$ term of the matrix $AB$ is different from the $(i,j)$ term of the matrix $BA$). Why the two definitions are equivalent and these movements are allowed?

Answer 1

The factors here are just ordinary numbers, not matrices. You can think of them as the entries of matrices, though.