Say we have a number field $K$ (specifically, an imaginary quadratic field) and a $2\times2$ matrix $\sigma=\pmatrix{a&c\\b&d}$ with elements $a,b,c,d\in\mathcal O_k$, the ring of integers of $K$. Generally, what is the significance of the norm of the ideal in $\mathcal O_k$ generated by the matrix elements, written $N(a,b,c,d)$, to the matrix $\sigma$?
For example, does the norm give us information about the eigenvalues of $\sigma$? If we write the norm as $N(\sigma)$, is the norm of a product $N(\sigma\tau)$ related to $N(\sigma)$ and $N(\tau)$ in a useful way, maybe involving determinants as well? Is $N$ or its square root a bona fide matrix norm?
I ask because in Vulakh (1994) "Reflections in extended Bianchi groups", the extended Bianchi group is represented by matrices for which $\det(\sigma)=\epsilon N(\sigma)$ where $\epsilon$ is a unit in $\mathcal O_k$. I don't really follow how one is led to write down that equation.