The inverse of a symmetric matrix (if it exists) is

Let A be an invertible symmetric matrix.

We have $A{A}^{-1}=A^{-1}A=I_n$

$\begin{array}{r}\Rightarrow {\left(A{A}^{-1}\right)}^{\mathrm{\prime }}={\left(A^{-1}A\right)}^{\mathrm{\prime }}={\left(I_n\right)}^{\mathrm{\prime }}\\ \Rightarrow {\left(A^{-1}\right)}^{\mathrm{\prime }}{A}^{\mathrm{\prime }}=A^{\mathrm{\prime }}{\left(A^{-1}\right)}^{\mathrm{\prime }}=I_n\\ \Rightarrow {\left(A^{-1}\right)}^{\mathrm{\prime }}A=A{\left(A^{-1}\right)}^{\mathrm{\prime }}=I_n\end{array}$

${\left(A^{-1}\right)}^{\mathrm{\prime }}=A^{-1}$    [inverse of a matrix is unique]