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