Chapter 2 - Matrices and Systems of Linear Equations - 2.6 The Inverse of a Square Matrix - Problems - Page 178: 34

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A is an $n \times n$ invertible symmetric matrix. Since A is symmetric, we will have $A^T=A$ and $(A^T)^{-1}=A^{-1}$ According to Theorem 2.6.10, $(A^T)^{-1}=(A^{-1})^T$ Hence here, $(A^{-1})^T=A^{-1}$ and $A^{-1}$ is symmetric