I am given a möbius transformation and asked to classify it.
$\frac{1}{3z}$
I know that a möbius transformation is generally of the form $\frac{az+b}{cz+d}$, so this möbius transformation is really $\frac{0z+1}{3z+0}$. From here, it is my understanding that I write a coefficient matrix of my möbius transformation and take the trace.
$$\begin{bmatrix} 0 & 1 \\ 3 & 0 \end{bmatrix}$$
In this case, the trace is $0$, which indicates that this möbius transformation is elliptical in nature.
I also know sometimes that you have to divide each element of the coefficient matrix by the square root of the determinant of the coefficient matrix before you find the trace.
In this case, dividing by the determinant will not affect the trace at all, but in other cases, how am I to differentiate when to divide by the determinant and when not to?
The trace of a normalized matrix (so its determinant is $1$) tells us the type of Mobius transformation it corresponds to. As you point out, normalizing doesn't affect the trace if the trace is $0$... but otherwise, it does affect the trace, so the answer is you have to normalize the matrix in all other cases.