A) Symmetric
B) Skew symmetric
C) Diagonal matrix
D) None of these
Correct Answer: A
Solution :
Let A be a symmetric matrix. Then \[A{{A}^{-1}}=I\Rightarrow {{(A{{A}^{-1}})}^{T}}=I\] \[\Rightarrow \] \[{{({{A}^{-1}})}^{T}}{{A}^{T}}=I\Rightarrow {{({{A}^{-1}})}^{T}}={{({{A}^{T}})}^{-1}}\] \[\Rightarrow \] \[{{({{A}^{-1}})}^{T}}={{(A)}^{-1}}\] , \[(\because \,\,{{A}^{T}}=A)\] \[\Rightarrow \] \[{{A}^{-1}}\] is a symmetric matrix.
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