Let ${(g{n}){n\\geq 1}}$ be a sequence of independent and identically distributed (i.i.d.) ${d\\times d}$ real random matrices. For ${n\\geq 1}$ set ${G_n = g_n \\ldots g_1}$. Given any starting point ${x=\\mathbb R v\\in\\mathbb{P}^{d-1}}$, consider the Markov chain ${X_n^x = \\mathbb R G_n v }$ on the projective space ${\\mathbb P^{d-1}}$ and define the norm cocycle by ${\\sigma(G_n, x)= \\log (|G_n v|/|v|)}$, for an arbitrary norm ${|\\cdot|}$ on $\\smash{\\mathbb R^{d}}$. Under suitable conditions we prove a Berry–Esseen-type theorem and an Edgeworth expansion for the couple ${(X_n^x, \\sigma(G_n, x))}$. These results are established using a brand new smoothing inequality on complex plane, the saddle point method and additional spectral gap properties of the transfer operator related to the Markov chain ${X_n^x}$. Cramér-type moderate deviation expansions as well as a local limit theorem with moderate deviations are proved for the couple ${(X_n^x, \\sigma(G_n, x))}$ with a target function ${\\varphi}$ on the Markov chain ${X_n^x}$.