Say $A$ and $B$ are symmetric, positive definite matrices. I've proved that 

$$\det(A+B) \ge \det(A) + \det(B)$$ 

in the case that $A$ and $B$ are two dimensional. Is this true in general for $n$-dimensional matrices? Is the following even true?

$$\det(A+B) \ge \det(A)$$ 

This would also be enough. Thanks.

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 by Marcus and Minc, Dover, 1992, P. 115 and also the previous MO thread). 