Suppose $A\in\Bbb{GL}_n(\Bbb R)$, (the space of all invertible matrices of order $n$), have integer entries.
If $\det(A)=\pm 1$ then obviously $A^{-1}$ have integer entries. Is the converse true, that is, if $A^{-1}$ have integer entries does it imply that $\det(A)=\pm 1$? Obviously for that $|\det(A)|=\gcd\{|A_{ij}|,1\leq i,j\leq n\}$ where $A_{ij}$ is the$(i,j)^{th}$ cofactor. But is it necessary for the $\gcd$ to be $1$ for a matrix to be invertible?