A) \[{{({{A}^{T}})}^{2}}={{A}^{T}}\]
B) \[{{({{A}^{T}})}^{2}}={{B}^{T}}\]
C) \[{{({{A}^{T}})}^{2}}={{({{A}^{-1}})}^{-1}}\]
D) None of the above
Correct Answer: A
Solution :
[a] Let A and B be two matrices such that AB = A and BA = B Now, consider AB = A Take transpose on both side \[{{(AB)}^{T}}={{A}^{T}}\] \[\Rightarrow {{A}^{T}}={{B}^{T}}.{{A}^{T}}...(1)\] Now, \[BA=B\] Take, transpose on both side \[{{(BA)}^{T}}={{B}^{T}}\] \[\Rightarrow {{B}^{T}}={{A}^{T}}.{{B}^{T}}....(2)\] Now, from equation (1) and (2). We have \[{{A}^{T}}=({{A}^{T}}.{{B}^{T}}){{A}^{T}}\] \[{{A}^{T}}={{A}^{T}}({{B}^{T}}{{A}^{T}})\] \[={{A}^{T}}{{(AB)}^{T}}(\because {{(AB)}^{T}}={{B}^{T}}={{B}^{T}}{{A}^{T}})\] \[={{A}^{T}}.{{A}^{T}}\] Thus, \[{{A}^{T}}={{({{A}^{T}})}^{2}}\]
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