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This work proposes a method to construct the Dirac operator in curved spacetime without introducing a vierbein (tetrad) or an independent spin connection, using only a matrix representation rooted in the basis structure of the four-dimensional gamma-matrix algebra. We introduce sixteen two-index gamma matrices realized as 256 × 256 matrices and embed the spacetime metric directly into matrix elements. In this framework, geometric operations such as covariantization, connection-like manipulations, and basis transformations are reduced to matrix products and trace operations, enabling a unified and transparent computational scheme. The spacetime dimension remains four; the number ``16'' labels the basis elements of the four-dimensional gamma-matrix algebra ((24 = 16). Based on an extended QED Lagrangian, the vertex rule, propagators, spin sums, and traces can be treated in a unified way, which facilitates automation. As validation, we consider Compton scattering, muon-pair production, Møller scattering, and Bhabha scattering. We show that off-diagonal components of the metric can induce characteristic angular dependences in differential cross sections, while the flat-spacetime limit reproduces standard QED results exactly. In a trial calculation with a toy metric containing off-diagonal components, a systematic deviation from the flat result appears near a scattering angle θ ≈ 90◦ when the coordinate angle is plotted directly, suggesting that metric-induced angular dependence could, in principle, serve as an observational indicator. These results indicate that the proposed matrix representation provides a practical algebraic tool to integrate the Dirac operator in a curved background and quantum electromagnetic processes into a single computational pipeline.