I've been doing some exercises in my workbook and every time I came across a problem involving a quadrupole moment it was always assumed that it was traceless without providing any proof (at least none that I saw) in the workbook. I mean it didn't really matter in solving the problem but it still annoys me that I don't know why it's traceless.
I've tried googling the proof to that but I couldn't find one. Each source always assumed that it's tracelss.
Now, am I just dumb and don't see it or what? Why is the quadrupole moment traceless?
Edit: The quadrupole moments I'm working with are defined as $$Q_{ij}=\sum_lq_l(3r_{il}r_{jl}-\|\vec{r_l}\|^2\delta_{ij}).$$ Now, I can't seem to figure out that it's traceless just from looking at that formula.