In the case of a sum $S_n$ of independent random variables, Bahadur and Rao and Petrov have established exact large deviation expansions for the probability $P(S_n geq nq)$ as $n$ goes tp $infty$. These milestone results have numerous applications in a variety of problems in pure and applied probability. Consider the product $G_{n}:=g_{n}...g_{1}$, where $(g_{n})_{n geq 1}$ is a sequence of i.i.d. $d times d$ real random matrices.The goal is to prove equivalent expansions for the norm $|G_n| $ and for the entries $G_n^{i,j}$.The asymptotics are expressed in terms of the eigenfunctions and invariant measures of the transfer operators related to the Markov chain representation of$log |G_n|$ and $log G_n^{i,j}$. In order to prove these results we develop the spectral gap theory for the scalar product of positive matrices. This is a joint work with Ion Grama and Quansheng Liu.