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Given an i.i.d. sequence $\{A_n(\omega)\}_{n\ge 1}$ of invertible matricesand a random matrix $B(\omega)$, we consider the random matrix sequencesinductively defined by $S_n(\omega) = A_n(\omega)S_{n-1}(\omega)$ and$T_n(\omega) =B(\sigma^{n-1}\omega)S_{n-1}(\omega)+A_n(\omega)T_{n-1}(\omega)$, and studyseveral limit theorems involving $T_n(\omega)$ as well as the asymptoticbehaviour of the action of $T_n(\omega)$ on the projective space and on theunit circle.

Some Limit Theorems Regarding Products of Random Matrices I: Directional Derivative of the Lyapunov Exponent