But, Xj = [Bji]Yi is still simply X=BY in matrix notation, because the contracted indices are the i in the vector with the second subscript (the i) in the matrix. This implies that the rows of B are multiplied by the column Y. In tensor notation, X=B[sup]T[/sup]Y would be represented by Xj=BijYi.
So, the upshot is that Bji and Bij are mutual transposes when they appear in the same expression; when they appear in different expressions they represent the same thing. What is important is the relative placements of indices which repeat (i.e., those which are contracted) in a single mulitiplicative expression.
Long answer… Bear with me. Don Roberto is correct, but it may be more useful to you this way:
The statement [b[sup]ij[/sup]] = [a[sub]ij[/sub]][sup]-1[/sup] means that b[sub]ij[/sub]a[sub]jk[/sub] = I[sub]ik[/sub], I being the identity matrix.
Given that, if I take my original equation
y[sub]i[/sub] = a[sub]ij[/sub]x[sub]j[/sub]
and multiply on the left by b[sub]ki[/sub], I get
b[sub]ki[/sub]y[sub]i[/sub] = b[sub]ki[/sub]a[sub]ij[/sub]x[sub]j[/sub] = I[sub]kj[/sub]x[sub]j[/sub] = x[sub]k[/sub]
This is exactly the same thing as x[sub]j[/sub] = b[sub]ji[/sub]y[sub]i[/sub], where we’ve just replaced “j” with “k,” which we’re free to do, as already discussed.
Oh, and a notational postscript. I’ve always seen B=A[sup]-1[/sup] expressed in this form via
B[sub]ij[/sub] = (A[sup]-1[/sup])[sub]ij[/sub], which is I think much less confusing.
Learning to switch between matrix notation and systems of equations is definitely one of the hardest parts of linear algebra. Stick with it; it’ll all make sense eventually.