Inverse of a symmetric positive definite matrix

We have $(A^{-1})^T = (A^T)^{-1}$ for any invertible matrix. It follows from this that if $A$ is invertible and symmetric $$(A^{-1})^T = (A^T)^{-1} = A^{-1}$$ so $A^{-1}$ is also symmetric. Further, if all eigenvalues of $A$ are positive, then $A^{-1}$ exists and all eigenvalues of $A^{-1}$ are positive since they are the reciprocals of the eigenvalues of $A$. Thus $A^{-1}$ is positive definite when $A$ is positive definite.

If A is positive definite matrix, then its eigenvalues are $\lambda_1, \dotsc, \lambda_n >0$ so,

\begin{equation} |A| = \prod_{i=1}^n \lambda_i > 0 \end{equation} and A is invertible. Moreover, eigenvalues of $A^{-1}$ are $\frac{1}{\lambda_i}>0$, hence $A^{-1}$ is positive definite. To see $A^{-1}$ is symmetric consider \begin{equation} A^{-1} = (A^T)^{-1}=(A^{-1})^T \end{equation}