  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
$n$.
