I am currently reading the book Baez & Muniain - "Gauge Fields, Knots, and Gravity". Chapter 2 of part II defines the holonomy on a bundle $E\to M$ with a gauge group $G$ and connection $D$. They end chapter 2 with the following:
If $\gamma$ is a loop based at $p\in M$, the holonomy $H(\gamma,D)$ is a linear map from $E_p$ to itself. In other words $H(\gamma,D)\in \text{End}(E)_p$. In this case, when we apply a gauge transformation $g$ to $D$, we have $$H(\gamma,D') = g(p)H(\gamma,D)g(p)^{-1},$$ so if we take the trace of $H(\gamma,D)$, we obtain a number that does not change under gauge transformations: $$\text{tr}(H(\gamma,D')) = \text{tr}(g(p)H(\gamma,D)g(p)^{-1}) = \text{tr}(H(\gamma,D)).$$ We therefore say that $\text{tr}(H(\gamma,D))$ is gauge invariant.
This trace quantity is apparently useful enough to be called a Wilson loop. Following the same method, I would imagine taking the determinant would also give us a gauge invariant quantity $$\det(H') = \det(gHg^{-1}) = \det(H)$$ and since $g,H$ are invertible linear maps, these would be nonzero.
Is the determinant of the holonomy around a loop gauge invariant (is my reasoning above valid)?
If so, is this a physically meaningful quantity? After a quick search, it seems like it is not as useful as the Wilson loop. Is there any specific reason for this?