Motivic classes of some classifying stacks

We prove that the class of the classifying stack $BPG{L}_n$ is the multiplicative inverse of the class of the projective linear group $PG{L}_n$ 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 $\{T\}=\{S\}\cdot \{PG{L}_n\}$ does not hold for all $PG{L}_n$-torsors $T\to S$, it holds for the universal $PG{L}_n$-torsors for said $n$.