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 $rank(A)=rank(A')$?