Question

If A is a non-singular symmetric matrix, write whether A⁻¹ is symmetric or skew-symmetric.

Answer 1

\[\text{ Let A be an invertible symmetric matrix . Then, }\]
\[ \left| A \right| \neq 0\text{ and }A^T = A\]
\[\text{ Now, }\left( A^{- 1} \right)^T = \left( A^T \right)^{- 1} \]
\[ \Rightarrow \left( A^{- 1} \right)^T = A^{- 1} [ \because A^T = A]\]
\[\text{ Thus, }A^{- 1}\text{ is symmetric matrix .}\]