> 
> Let $B$ and $C$ be $n \times n$ Hermitian matrices, with $B$ positive definite and $C$ positive semidefinite.
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> 1. Show that $B+C$ is positive definite
> 2. Show that $\det(B) \leq \det(B+C)$. What is the equality case?
> 3. Show that $B^{-1}-(B+C)^{-1}$ is positive semidefinite.
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I proved (1). And for (2), I tried using Cholesky decomposition but does not gain anything.

Any hints, ideas? Thanks.