Motivated primarily by the study of large deviations of multitype branching processes in random environments, we first establish, for products of independent and identically distributed random positive matrices $(M_n)_{n \in \mathbb Z}$, a Perron-Frobenius type theorem under the Cramér type changed measure, the stable and mixing convergence of the direction of the random walk $x M_0 \cdots M_n$ (with $x \in \mathbb R_+^d$) as $n \to \infty$, under both the initial probability and the changed measure. We also determine the exact growth rate of the moments of the vector norm $\| x M_0 \cdots M_n\|$, the entrywise $L^1$ matrix norm $\| M_0 \cdots M_n\|_{1,1}$, and the scalar product $\langle x M_0 \cdots M_n, y \rangle $ for $x, y \in \mathbb R_+^d$ with unit norm. As applications, we derive precise large deviation results for the population size $\| Z_n \|$ of $n$-th generation, for a $d$-type branching process $Z_n=(Z_n(1), \cdots, Z_n(d))$ in an independent and identically distributed random environment, by giving an equivalence of the large deviation probability $ \mathbb P [\| Z_n \| \geq e^{nq}]$, for $q>0$ in a suitable range. Additionally, we obtain precise estimation of the moments of $\|Z_n \|$ and those of the $j$-type population size $Z_n (j)$.