Non-PSD matrix with two different PSD submatrixes Let $A$ be an $n \times n$. Define $A_{-i}$ to be the matrix $A$ without the $i$-th column and row. For instance
$$
A=
\begin{pmatrix}
1 & 2 & 3\\
4 & 5 & 6\\
7 & 8 & 9
\end{pmatrix}
\implies
A_{-2}=
\begin{pmatrix}
1 & 3\\
7 & 9
\end{pmatrix}
$$
Is it possible to find $S$ symmetric and not positive semidefinite such that there exist $i_1\not= i_2$ with $A_{-i_1}, A_{-i_2}$ which are positive semidefinite?

$$
\left(
\begin{array}{ccc}
5 & -3 & -3 \\
-3 & 5 & -3 \\
-3 & -3 & 5 \\
\end{array}
\right)
$$
$$
\left(
\begin{array}{ccc}
7 & -4 & -5 \\
-4 & 8 & -6 \\
-5 & -6 & 9 \\
\end{array}
\right)
$$
