This paper analyzes how difficult it is, from a computational viewpoint, to decide whether two orthogonal projection operators commute. It defines the problem precisely, including how the input matrices are represented and how computational cost is measured. The study proves that this decision can always be made in deterministic polynomial time by a straightforward algorithm that multiplies and compares the matrices. The author also examines the bit-level cost of computation and shows it remains polynomial in both matrix size and numerical precision. Beyond the algorithm, the paper connects the result to operator theory by describing equivalent ways to detect commutativity, improving both conceptual understanding and numerical practicality.