I'm searching for rather specific counter-example.


Some notation: $A(\alpha|\beta)$ is the sub matrix of $A$ with with rows $\alpha$ and columns $\beta$. $\textrm{det } A(\alpha|\alpha) =: \textrm{det } A(\alpha)$ are the principal minors of $A$. We define $\textrm{det } A(\emptyset)=1$. A matrix $A$ is irreducible if there is no permutation matrix $P$ so that


$$ P^{-1} A P = \begin{bmatrix} E & G \\ 0 & F \end{bmatrix} $$


where $E$ and $F$ are square.


Is there a invertible, irreducible matrix $A \in \mathbb{C}^{4 \times 4}$ with $\textrm{det } A(\alpha) = \textrm{det } ((\overline{A}^{-1})(\alpha))$ for all $\alpha \subseteq \{1,2,3,4\}$ and $\textrm{rank } A([1,2],[3,4]) = 1$?



Example for Reciprocal Principal Minors
The matrix $$A=\frac15\left(
\begin{array}{cccc}
 -1 & 4 & -2 & -2 \\
 -4 & 1 & 2 & 2 \\
 2 & 2 & -1 & 4 \\
 -2 & -2 & -4 & 1 \\
\end{array}
\right)$$is orthogonal, i. e. $A=\overline{A}^{-1}$.


