Motivic classes of some classifying stacks

We prove that the class of the classifying stack BPGLn is the multiplicative inverse of the class of the projective linear group PGLn in the Grothendieck ring of stacks K0(Stackk) 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}·{PGLn} does not hold for all PGLn‐torsors T→S, it holds for the universal PGLn‐torsors for said n.