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Let $(g_{n})_{n\geq 1}$ be a sequence of independent and identicallydistributed positive random $d\times d$ matrices and consider the matrixproduct $G_n: = g_n \ldots g_1$. Under suitable conditions, we establish theBerry-Esseen bounds on the rate of convergence in the central limit theorem andmoderate deviation expansions of Cram\'er type, for the matrix norm $\| G_n \|$of $G_n$, for its $(i,j)$-th entry $G_n^{i,j}$, and the and for its spectralradius $\rho(G_n)$.

Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices