Given a matrix with natural numbers $\geq 0$ as entries and having determinant equal to one and positive diagonal entries. Is it the Cartan matrix of a finite dimensional algebra of finite global dimension? If not, can the matrices having this property be classified?

The Cartan matrix of a finite dimensional algebra is defined as the matrix having entries $c_{i,j}:=$dimension of $(Hom_{A}(P_i,P_j))$ over the divison ring $D_i$, where $D_i:=End_A(S_i)$. Here $P_i$ are the indecomposable projective modules and $S_i$ the simple modules which are the top of $P_i$.

We can look also at small cases for $n=1,2,3,...$ . I begin: For $n=1$ all matrices can be realised, since the only one is $[1]$, which is realised by any division ring.