TITLE: Matrix inequality after taking inverse
QUESTION [9 upvotes]: Let A and B be Positive definite Matrices with $ A\leq B$ in the sense that $B-A$ is positive definite. Is it true that $A^{-1} \geq B^{-1} $?

REPLY [5 votes]: Consider $B-A\geq 0$ using Schur complement, this is equivalent to
$$\begin{bmatrix}B&I\\I&A^{-1}\end{bmatrix}\geq 0, \quad B>0$$
Since $A^{-1}>0$, now apply the Schur complement one more time to obtain
$$A^{-1}-I(B)^{-1}I=A^{-1}-B^{-1}\geq  0$$
therefore we have $A^{-1}\geq B^{-1}$.