If $A$ is a positive definite matrix and $\text{det}B=0$, then $\text{det}(A+B)$ is positive?

Question

If $A$ is a positive definite matrix and a symmetric matrix $B$ satisfies $\text{det}B=0$, then $\text{det}(A+B)$ is positive?

I tired to find a counterexample for this, but I couldn't.

Answer 1

Counterexample:

$$\mathbf A=\begin{pmatrix}-2&1\\1&-7\end{pmatrix}\\\mathbf B=\begin{pmatrix}1&3\\3&9\end{pmatrix}$$ $\det(\mathbf{A})>0$ and $\det(\mathbf{B})=0$, but $\det(\mathbf{A}+\mathbf{B})=(-1)\cdot 2-4\cdot 4=-18<0$.