Q:

A question about the determinant of matrices with integer entries

Motivated by some Physics backgrounds, let's consider the following group $GL_n(\mathbb Z)$ which consists of matrices satisfying some conditions.   
Let $M_n(\mathbb Z)$ be the set(not a group) of $n\times n$ matrices $A$ with integer entries and $\det A\neq 0$. Then the set $GL_n(\mathbb Z)$ is defined as:
$GL_n(\mathbb Z)=\left \{ A: A\in M_n(\mathbb Z),\ {A}^{-1}\in M_n(\mathbb Z) \right \}$. It's easy to verify that $GL_n(\mathbb Z)$ is a group.  
Now my question is as follows: $\forall A\in GL_n(\mathbb Z)$, whether $\det(A)=1$ or $-1$?  I have checked that it's true for $n=1,2$ cases, but for $n\geq 3$ cases I can't prove it.  Does this kind of group $GL_n(\mathbb Z)$ is well studied in mathematics?  Or maybe this question makes no sense. 
Thanks in advance.  

A:

$1=\det(AA^{-1})=\det(A)\det(A^{-1})$.  But $A,A^{-1}$ have integer entries, so have integer determinants.   Thus the determinants are either both 1 or both $-1$.

