Experts argue that public key cryptography systems based on commutative groups (abelian groups) are quite vulnerable to quantum algorithms attacks. For this reason, experts have developed a non-commutative group-based public-key cryptosystem. In this study, we propose a public-key cryptosystem using the non-commutative property of the function composition of the matrix. The results of this study indicate that the proposed public-key cryptosystem can work well. Furthermore, the proposed cryptosystem is also safe from various attempted mathematical attacks on the key exchange process at the sender and receiver of the message.