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Question
If A is a non-singular symmetric matrix, write whether A−1 is symmetric or skew-symmetric.
Solution
\[\text{ Let A be an invertible symmetric matrix . Then, }\]
\[ \left| A \right| \neq 0\text{ and }A^T = A\]
\[\text{ Now, }\left( A^{- 1} \right)^T = \left( A^T \right)^{- 1} \]
\[ \Rightarrow \left( A^{- 1} \right)^T = A^{- 1} [ \because A^T = A]\]
\[\text{ Thus, }A^{- 1}\text{ is symmetric matrix .}\]
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