Inverse of integer matrix with determinant $\pm 1$

Question

Suppose we have a matrix $A = (a_{ij}) \in \operatorname{GL}_n(\mathbb{R})$ with $a_{ij} \in \mathbb{Z}$. I need to show that $A^{-1}$ has entries in $\mathbb{Z}$ if and only if $\det(A) = \pm 1$.

Answer 1

Using Cramer's Rule to obtain the inverse of $A$, we can see that $A^{-1}$ has entries in $\Bbb Z$, if $A$ has entries in $\Bbb Z$ and $\det(A)=\pm 1$. For the other direction of the proof, we use $$\det(A)\det(A^{-1})=\det(AA^{-1})=\det(I)=1$$ If $A$ and $A^{-1}$ have entries in $\Bbb Z$, then $\det(A)\in\Bbb Z$ and $\det(A^{-1})\in \Bbb Z$, and the equation above can hold only if $\det(A)=\det(A^{-1})=\pm 1$.