Let $A,B\in M_{n}(\mathbb{R})$ be symmetric, with $A>0$ and $B>0$.
I need to prove that $\det (A+B)>\max (\det(A), \det(B))$.
I want to use Sylvester theorem of having a matrix $D$ so that $D=\operatorname{diag}(1,1,\ldots, 1,-1,-1, \ldots-1,0,0,\ldots,0)$.
Do I need to use it? How do I use it here?
Thank you.