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3 years ago

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

Mathematics

1 answer:

Lisa [10]3 years ago

3 0

Answer:

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

Step-by-step 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.

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

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$\\ {\large{\textsf{\textbf{\underline{\underline{ Question \: :}}}}}} \\$

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$\qquad\rule{120pt}{2pt}$

$\\ {\large{\textsf{\textbf{\underline{\underline{ Solution \: :}}}}}} \\$

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