Is there a general rule to identify all $n\times n$ matrices $A$ consisting only of entries $0,1$ such that the matrix has an odd determinant (and hence invertible) and is not the inverse of itself?
I know specific examples of such $A$. Like ones along the diagonal and above the diagonal and zeros below the diagonal is invertible and $A^{-1}\neq A$ but am wondering if we can identify all such $A$?