  In this paper we study a 2-type linear-fractional branching process in
varying environment with asymptotically constant mean matrices. Let $\nu$ be
the extinction time and for $k\ge1$ let $M_k$ be the mean matrix of offspring
distribution of individuals of the $(k-1)$-th generation. Under certain
conditions, we show that $P(\nu=n)$ and $P(\nu>n)$ are asymptotically
equivalent to some functions of products of spectral radii of the mean
matrices. This paper complements a former result [arXiv: 2007.07840] which
requires in addition a condition $\forall k\ge1,\rm{det}(M_k)<-\varepsilon$ for
some $\varepsilon>0.$ Such a condition excludes a large class of mean matrices.
As byproducts, we also get some results on asymptotics of products of
nonhomogeneous matrices which have their own interests.

Probability