When the inverse of a matrix with integer entries also has integer entries

If $A$ and $A^{-1}$ have integer entries, then $$\det(A)\det(A^{-1})=\det(AA^{-1})=\det(I)=1$$ and as $\det(A)$ and $\det(A^{-1})$ are integers, then $\det(A)=\det(A^{-1})=\pm1$.