Question

Prove the theorem (AB )^T= B^T. A^T

Answer 1

`(AB)^T = B^T.A^T (Proved)`

Explanation:

Given (AB )^T= B^T. A^T;

To prove this expression, we need to apply multiplication law, power law and division law of indices respectively, as shown below.

`(AB)^T = B^T.A^T\\\\Start, from \ Right \ hand \ side\\\\B^T.A^T = (B^T.A^T)/(A^T).(B^T.A^T)/(B^T) (multiply \ through) \\\\ = (A^(2T).B^(2T))/(A^T.B^T) \\\\=((AB)^(2T))/((AB)^T) \ \ (factor \ out \ the power)\\\\= (AB)^(2T-T) \ (apply \ division \ law \ of \ indices; \ (x^a)/(x^b) = x^(a-b))\\\\= (AB)^T \ (Proved)`