Quantum matrix pairs

The notion of quantum matrix pairs is defined. These are pairs of matrices with noncommuting entries, which have the same pattern of internal relation s, q-commute with each other under matrix multiplication, and are such that products of powers of the matrices obey the same pattern of internal relat ions as the original pair. Such matrices appear in an approach by the autho rs to quantizing gravity in 2 space and 1 time dimensions with negative cos mological constant on the torus. Explicit examples and transformations whic h generate new pairs from a given pair are presented.