Inverse of a sum of positive definite matrices Let $A,B$ be symmetric positive definite matrices. Let $A^{-1} = LL^T$ (Cholesky decomposition, $L$ is lower-triangular). I think the following identities are true, but I haven't found them online:
$$
(A+B)^{-1} = L(I+L^TBL)^{-1}L^T \\
|A+B| = |A| \, |I+L^T B L|
$$
Are they correct? And if so, how do you show they're true? I suppose they can be derived from the Woodbury formula and the matrix determinant lemma, but I tried and I couldn't prove it...

Hint:
notice that: 
$$L^{-1}(A+B)^{-1}L^{-T}=(I+L^TBL)^{-1}$$
Also
$$L^{-1}(A+B)^{-1}L^{-T}=(L^T(A+B)L)^{-1}$$
