Why was noncommutative multiplication important to matrix mechanics?

In 1925, Werner Heisenberg sought to develop a theory of quantum mechanics based exclusively on observable quantities, such as the spectrum intensities of an electron. His resulting formalism relied on noncommutative multiplication, a mathematical property where the order of operations affects the outcome.1

This approach was significant because it provided the necessary structure to describe the kinematics and mechanical relations of atomic systems without relying on classical, visualizable orbits. Max Born and Pascual Jordan recognized that this noncommutative behavior could be expressed using matrix algebra, leading to the development of matrix mechanics. This framework established a firm conceptual foundation for quantum theory, moving away from the unsatisfactory ad hoc mixtures of classical and quantum ideas that had previously dominated the field.12