This paper introduces a new algorithm for square matrix multiplication called Alpha-Beta Matrix Multiplication. Unlike traditional methods that recalculate each element independently, this approach exploits value reuse opportunities to reduce computational complexity. By transforming the matrix multiplication problem into a form that allows for precomputation and reuse of intermediate values, we achieve consistent performance improvements over the standard three-loop method. Empirical testing demonstrates that our algorithm achieves an average speedup of 1.28$\times$ across various matrix sizes, with minimal additional memory requirements.