A) \[{{A}^{-n}}{{B}^{n}}{{A}^{n}}\]
B) \[{{A}^{n}}{{B}^{n}}{{A}^{-n}}\]
C) \[{{A}^{-1}}{{B}^{n}}A\]
D) \[n({{A}^{-1}}BA)\]
Correct Answer: C
Solution :
\[{{({{A}^{-1}}BA)}^{2}}=({{A}^{-1}}BA)({{A}^{-1}}BA)\] \[={{A}^{-1}}B(A{{A}^{-1}})BA\] \[={{A}^{-1}}BIBA={{A}^{-1}}{{B}^{2}}A\] \[{{({{A}^{-1}}BA)}^{3}}=({{A}^{-1}}{{B}^{2}}A)({{A}^{-1}}BA)\] \[={{A}^{-1}}{{B}^{2}}(A{{A}^{-1}})BA\] \[={{A}^{-1}}{{B}^{3}}A\]and so on \[\therefore \]\[{{({{A}^{-1}}BA)}^{n}}={{A}^{-1}}{{B}^{n}}A\]You need to login to perform this action.
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