The inequality 
$$\det(A+B)\geq \det A +\det B$$
is implied by the Minkowski determinant theorem
$$(\det(A+B))^{1/n}\geq (\det A)^{1/n}+(\det B)^{1/n}$$
which holds true for any non-negative $n\times n$ Hermitian matrices $A$ and $B$. The latter inequality is equivalent to the fact that the function $A\mapsto(\det A )^{1/n}$ is concave on the set of $n\times n$ non-negative Hermitian matrices (see e.g. [*A Survey of Matrix Theory and Matrix Inequalities*][1] by Marcus and Minc, Dover, 1992, P. 115). 


  [1]: http://books.google.com/books?id=hLHKwSNqLOcC&printsec=frontcover&dq=marcus++matrix&hl=en&ei=6hXVTfi-M4Gg-wag_5TvCw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDYQ6AEwAA#v=onepage&q&f=false