Question

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

Answer 1

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