I'm reading the text Representations of Finite Groups by Nagao and Tsushima. In an exercise, it asked to prove $\det C > 0,$ where $C$ is the Cartan matrix of a group ring $RG$ and $R$ is a complete discrete valuation ring.
I know that $C = D' D$ (where $D$ is the decomposition matrix of $RG$ and $D'$ is the transpose of $D$) and $D$ has full column rank. If $D$ is a square matrix, then it's obvious that $\det C = (\det D)^2 > 0,$ but the problem is that $D$ is usually not a square matrix.
If you don't have a background of group representation theory, my question is
if $C=D'D$ $(C$ and $D$ are integer matrices) and $D$ has full column rank, does $\det C > 0$ hold?