  We study the asymptotic behaviour of the survival probability of a multi-type
branching processes in random environment. The class of processes we consider
corresponds, in the one-dimensional situation, to the intermediately
subcritical case. We show under rather general assumptions on the form of the
offspring generating functions of particles that the probability of survival up
to generation $n$ of the process initiated at moment zero by a single particle
of any type is of order $\lambda ^{n}n^{-1/2}$ for large $n,$ where $\lambda
\in (0,1)$ is a constant specified by the Lyapunov exponent of the mean
matrices of the process.
