A problem in fractal geometry requires me to consider matrix semigroups with the following curious property. For a $d \times d$ real matrix $A$ let $\lambda_1(A) \geq \lambda_2(A) \geq \cdots \geq \lambda_d(A)$ denote the absolute values of the eigenvalues of $A$. I am interested in semigroups $\mathcal{S} \subset GL(d,\mathbb{R})$ with the property that for some fixed $t \in (0,1)$ depending only on $\mathcal{S}$, $$\lambda_1(AB)\lambda_2(AB)^t = \lambda_1(A)\lambda_2(A)^t\lambda_1(B)\lambda_2(B)^t$$ for all $A,B \in \mathcal{S}$.

Clearly a sufficient condition for the above property is that the absolute eigenvalues $\lambda_1$, $\lambda_2$ are separately multiplicative in $\mathcal{S}$: that is, $\lambda_i(AB)=\lambda_i(A)\lambda_i(B)$ for all $A,B \in\mathcal{S}$ and $i=1,2$. My question is: is this condition also necessary? Is there any other mechanism which can lead to the above property being satisfied?