Is the trace of the matrix obtained by subtracting two positive definite matrices smaller than the trace of the matrix being subtracted?

Question

Let $A$, $B$ be symmetric positive definite matrices. If $C=A-B$, is it true that $$ \operatorname{trace}\left(C^{-1}\right) > \operatorname{trace}\left(A^{-1}\right)\:?$$

Answer 1

Are you assuming that $C$ is also positive definite?

If yes, then this follows from the fact that $A\succ C$ so $C^{-1}\succ A^{-1}$, as proved here. (Here $A\succ C$ means that $A-C$ is positive definite.)

If not, then $A=I,B=2I$ is a counterexample.