Multiplication is not commutative. You used "from the right" on the left and "from the left" on the right. No good.

You did it again! How did that A^(-1) get on the right of that '2A'?BAA^-1 = A^-1(2A-I)

B = (2A)A^-1 - A^-1(I)

Ack! I'm pretty sure you mean 2I, not just 2.B = (2A)A^-1 - A^-1(I)

B = 2 - A^-1

Okay, that was some lovely algebra. Did you answer the question?