Motivic classes of some classifying stacks

Abstract: We prove that the class of the classifying stack $B PGL_n$ is the multiplicative inverse of the class of the projective linear group $PGL_n$ in the Grothendieck ring of stacks for $n = 2$ and $n = 3$ under mild conditions on the base field $k$. In particular, although it is known that the multiplicativity relation ${T} = {S}\cdot{PGL_n}$ does not hold for all $PGL_n$-torsors $T \to S$, it holds for the universal $PGL_n$-torsors for said $n$.