<s>[INST] Positive definiteness of difference of inverse matrices Let $A$ and $B$ be two $n \times n$ symmetric and positive definite matrices. 
If $A \prec B$, then is it true that $B^{-1} \prec A^{-1}$?
Here, $A \prec B$ means that $B-A$ is positive definite.
 [/INST] For ease of notation I'll use $\ge$ instead of $\succeq$ and so on.
Lemma 1: $A \le B \Rightarrow C^T A C \le C^T B C$ for any conformable matrix $C.$
Proof. We have $x^TC^T(B-A)C x = (Cx)^T(B-A)C x \ge 0$ for any conformable vector $x$ so that $C^T(B-A)C \ge 0.$
Lemma 2: $I \preceq B \Rightarrow$ $B$ is invertible and $B^{-1} \preceq I.$
Proof.

*

*Since $B$ is symmetric we can find an orthogonal matrix $G$ and a diagonal matrix $D$ such that $B=GDG^T.$ Hence $I = G^T I G \le G^T B G = D.$ Thus all eigenvalues of $B$ are $\ge 1 >0,$ hence $B$ is invertible.

*Now write $B^{-1} = B^{-1/2}IB^{-1/2} \le B^{-1/2}BB^{-1/2} = I,$ where $B^\alpha := GD^{\alpha}G^T.$
Proposition: $0 < A \le B \Rightarrow$ $B$ invertible and $B^{-1} \le A^{-1}.$
Proof.
\begin{align*}
A\le B &\Rightarrow B-A \ge 0 \\
       &\Rightarrow A^{-1/2}(B-A)A^{-1/2} \ge 0\\
       &\Rightarrow A^{-1/2}BA^{-1/2} \ge I\\
       &\Rightarrow A^{1/2}B^{-1}A^{1/2} \le I,\\
\end{align*}
hence
\begin{align*}
B^{-1} &= A^{-1/2}A^{1/2}B^{-1}A^{1/2}A^{-1/2}\\
       &= A^{-1/2}(A^{-1/2}B^{-1}A^{-1/2})^{-1}A^{-1/2}\\
       &\le A^{-1/2}IA^{-1/2}\\
       &= A^{-1}.
\end{align*}
</s>