First version: Let $A$ and $B$ be (complex) Hankel matrix. Is it true that $\det (A+B)\neq 0$ if $\det A=0$ and $\det B\neq0$? **No**.


Reformulating: For which $B$ is it true that $\det (A+B)\neq 0$ if $\det A=0$?



Determinant of a sum of two Hankel matrices
No, let
$$
B=\left(\begin{matrix}
0 & 1\\
1 & 1\\
\end{matrix}\right)
$$
$$
A=\left(\begin{matrix}
1 & 0\\
0 & 0\\
\end{matrix}\right)
$$


