Combinatorial problems serve as an important resource for developing practical public key cryptosystems and several combinatorial cryptosystems have been proposed in the cryptographic community. In this paper, a combinatorial public key cryptosystem is proposed. The security of the proposed cryptosystem is dependent on a combinatorial problem involving matrices. The system features fast encryption and decryption. However, the system also suffers from some drawbacks. The ciphertext expansion is relatively large and the key sizes are somewhat larger than that of RSA. The security of the system is carefully examined by illustrating the computational infeasibilities of some attacks on the system.