General linear group over integers I'm trying to get my head around group theory as I've never studied it before.
As far as the general linear group, I think I've ascertained that it's a group of matrices and so the 4 axioms hold? 
The question I'm trying to figure out is why $(GL_n(\mathbb{Z}),\cdot)$ does not form a group. 
I think I read somewhere that it's because it doesn't have an inverse and I understand why this would not be a group, but I don't understand why it wouldn't have an inverse. 

$\DeclareMathOperator{\GL}{GL}\GL_n(\mathbf Z)$ is a multiplicative group, by definition: it is the set of invertible matrices with coefficients in $\mathbf Z$. 
The problem is that it's not what you seem to think – the set of matrices with a non-zero determinant. In general, for any (commutative) ring $A$, $\GL_n(A)$ is the set
/group of invertible matrices, i. e. the matrices with determinant invertible in $A$.
In the case $A=\mathbf Z$, this means the matrix has determinant $\pm1$.
