In recent years, Random Network Coding (RNC) has emerged as a promising solution for efficient Peer-to-Peer (P2P) video multicasting over the Internet. This probably refers to this fact that RNC noticeably increases the error resiliency and throughput of the network. However, high transmission overhead arising from sending large coefficients vector as header has been the most important challenge of the RNC. Moreover, due to employing the Gauss-Jordan elimination method, considerable computational complexity can be imposed on peers in decoding the encoded blocks and checking linear dependency among the coefficients vectors. In order to address these challenges, this study introduces MATIN which is a random network coding based framework for efficient P2P video streaming. The MATIN includes a novel coefficients matrix generation method so that there is no linear dependency in the generated coefficients matrix. Using the proposed framework, each peer encapsulates one instead of n coefficients entries into the generated encoded packet which results in very low transmission overhead. It is also possible to obtain the inverted coefficients matrix using a bit number of simple arithmetic operations. In this regard, peers sustain very low computational complexities. As a result, the MATIN permits random network coding to be more efficient in P2P video streaming systems. The results obtained from simulation using OMNET++ show that it substantially outperforms the RNC which uses the Gauss-Jordan elimination method by providing better video quality on peers in terms of the four important performance metrics including video distortion, dependency distortion, End-to-End delay and Initial Startup delay.