Show that $det \left( {A+B} \right) \geq det\left( {A} \right)$

Question

Let $A$ is a positive define symmetric matrix,and $B$ is a semi-positive definite symmetric matrix. Does $$det \left( {A+B} \right) \geq det\left( {A} \right)$$ hold? Could someone help me with this, thank you.

Answer 1

It turns out I have answered a similar question before, no wonder I have a feeling of "déjà vu".
Look at answers in that question if you find the answer below too terse.

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Since $A$ is positive definite, it has a positive definite invertible square root $S = A^{1/2}$. We have the breakdown $$\det(A+B) = \det(A)\det(I+ C)$$ where $C = S^{-1/2}BS^{-1/2}$.

Since $B$ is positive semi-definite and symmetric, so do $C$. If we choose a basis to diagonalize $C$, we find its eigenvalues are all non-negative. This implies $\det(I+C) \ge 1$ and we are done.