Consider the definition from wikipedia:
Definition 1: A matrix $M\in\mathbb R^{n\times n}$ is said to bre positive semi-definite if $x^TMx \ge 0$ for all $x \in \mathbb R^n$.
We use the following property from wikipedia again
Fact 1: A matrix $M$ is positive-definite then $\exists B$ nonsingular such that $M = B^HB$.
Solution: Write $A = M^TM$, $M$ nonsingular (fact 1). Define $N = (M,M^{-T})$. We have $$\left[ \begin{matrix} A & I \\ I & A^{-1} \end{matrix} \right] = \left[ \begin{matrix} M^TM & M^TM^{-T} \\ M^{-1}M & M^{-1}M^{-T} \end{matrix} \right] = N^TN \ge 0.$$
I am not sure how it follows that $N^TN \ge 0$. Any explanation is greatly appreciated!