rt.representation theory - When is a generalized Cartan matrix invertible?

Let $A=(a_{ij})$ be a generalized Cartan matrix, i.e. $a_{ij} in Z, a_{ii}=2$, $a_{ij}leq 0$ for $i neq j$ and $a_{ij}=0$ iff $a_{ji}=0.$
If $A$ is classical Cartan matrix or hyperbolic, it is known that $A$ is invertible, while if $A$ is affine it has a 1-dimensional kernel.

What is known about general (indecomposable) $A$ of indefinite type?

EDIT: If $A in Z^n$ is invertible, then one clearly can find $v,w in Z^n$
such that $$A':=
begin{pmatrix} A & v \
w^t & 2 end{pmatrix}$$

is again an indecomposable gCM. So the question should be: Given $A$ invertible, how do you produce $A'$ such that $A$ is the upper-left corner of $A'$ and $rank(A)=rank(A')$?