Problem: Show for two positive definite matrices $A$ and $B$, if $A \succeq B$ then $B^{-1} \succeq A^{-1}$
Definition: $A \succeq B$ if $A-B$ is positive semidefinite.
I've been trying to use the adjunct matrices since $A^{-1}=\frac{1}{det(A)}adj(A)$ so from the $\frac{1}{det(A)}$ it seems sort of obvious that if $A \succeq B$ then $B^{-1} \succeq A^{-1}$. However, I have no idea beyond this how to use the positive definite nature of $A$ and $B$ or definition of $A \succeq B$ to prove the statement. All I know is that if a matrix $A$ is positive definite then so is $A^{-1}$.
Any help would be appreciated!
