A) \[{{A}^{n}}{{B}^{n}}{{A}^{-n}}\]
B) \[{{A}^{-1}}{{B}^{n}}A\]
C) \[{{A}^{-n}}{{B}^{n}}{{A}^{n}}\]
D) None of these
Correct Answer: B
Solution :
Let A and, Bare square matrices of same order and A is non-singular. For positive integer n, consider \[{{({{A}^{-1}}BA)}^{n}}=({{A}^{-1}}BA)({{A}^{-1}}BA)({{A}^{-1}}BA)\underbrace{.........}_{\text{n}\,\text{times}}({{A}^{-1}}BA)\]\[={{A}^{-1}}B(A{{A}^{-1}})B(A{{A}^{-1}})B(A{{A}^{-1}})...(A{{A}^{-1}})BA\] \[={{A}^{-1}}Bl\,Bl\,Bl...lBA\] \[\because \{A{{A}^{-1}}=Bl\}\] \[={{A}^{-1}}\underbrace{B.B.B....}_{\text{n}\,\text{times}}BA={{A}^{-1}}{{B}^{n}}A\] \[\Rightarrow \] \[{{({{A}^{-1}}BA)}^{n}}={{A}^{-1}}{{B}^{n}}A\]You need to login to perform this action.
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