Given an invertible matrix $A \in \mathbb{R}^{n \times n}$, and index set $\langle n\rangle = \{ 1, \dots, n \}$, and the submatrix $A(\alpha)$ with the columns and rows of $A$ with indices $\alpha \subset \langle n \rangle$.

Is there are characterisation of all matrices $A$ with

$\textrm{det } A(\alpha) = \textrm{det } A^{-1}(\alpha) \qquad \forall \alpha \subset \langle n \rangle \qquad ?$

Example: Sufficient conditions are involutory ($A^{-1} = A$) or orthogonal ($A^{-1} = A^T$) matrices.