The growing need of safe data transmission in the contemporary digital setting, such as cloud-based, Internet of Things (IoT), and distributed systems, has revealed the weaknesses of the traditional cryptography methods, especially in the form of new quantum threats. Traditional approaches founded on number-theoretic conjectures are now facing challenges, and more multifaceted and scalable mathematical theories need to be pursued. This work tackles this problem by suggesting a new cryptographic transformation model that is based on matrix group representations, which are defined on finite fields. The main aim of the work is to construct a mathematically sound and implementation-friendly encryption scheme that increases the security level by high-dimensional, non-commutative transformations. The methodology consists of a representation mapping of plaintext data in the form of vectors and a series of reversible matrix transformations, based on group elements. The system combines the process of key generation, encryption, and decryption in one algebraic framework and is executed in Python (NumPy, SciPy) and MATLAB to perform efficient operations on matrices. The reported experimental results indicate ciphertext entropy of 7.98 bits per byte and an avalanche effect of 52.6%, compared with 7.95 bits and 50.1%, respectively, for the AES baseline. These preliminary findings suggest favorable statistical randomness and diffusion characteristics, although reproducibility requires further documentation of the experimental protocol and independent validation of the reported measurements. Comparative analysis to AES indicates that the proposed scheme has a slightly higher computational overhead, but provides a better key space, structural complexity, and resistance to attacks. Scalability tests ensure that the system works effectively with a moderate size of matrices as well as various data sets, such as text and picture data. Overall, the proposed framework provides safe, scalable and versatile solution to next generation cryptography usage, particularly high security and post-quantum settings.