I have a question for Cartan matrix.
In the quiver representation theory, the Cartan matrix is defined as follows:
Let $Q$ be a quiver of type $A_n$ without oriented cycles Let $c_{ij}$ be the number of paths from $j$ to $i$. The matrix $C=(c_{ij})$ is called the Cartan matrix.
In my book[Ralf Schiffler, Quiver Representations, Springer] at page 74, the author has proved that $C$ is invetible by saying as follows:
Since $Q$ has no oriented cycles, we can renumber vertices to be $i\le j$ if there is a path from $j$ to $i$ in $Q$. Then the revised matrix is upper triangular with all diagonals 1, so $C$ is inveritible.
I have a question at this point. If we renumber our quiver $Q$, then the quiver is changed, so that we may get a new quiver. So, of course, the corresponding Cartan matrix of the new quiver is invertible upper triangular, but I cannot check whether the relation between the old and new ones is or not. More precisely, if the new Cartan matrix is invertible, then so is the old Cartan matrix, why?
Thanks in advance.