In the existing key pre-distribution scheme suitable for low power and resource sensor nodes, mutual authentication is not allowed and the security tolerance of network is low. In this paper, we propose a new key pre-distribution scheme guaranteeing that any two sensor nodes can find a shared key between themselves. The basic idea of the paper is to combine matrix decomposition technique and polynomial-based key pre- distribution approach. The analysis indicates the proposed scheme allows almost 100% connectivity regardless of the number of keys and it enhances security by node-to-node mutual authentication. On the other hand, introducing polynomial-based key pre-distribution, our scheme greatly improves the resilience against node capture.