  We show that the K_1 group of a C*-algebra $A$ can be defined as homotopy
classes of pairs, called balanced, of not necessarily unitary matrices over $A$
that have equal defects from being unitary. We also consider pairs of order
zero pseudodifferential operators, not necessarily elliptic, with symbols being
a balanced pair. A relative index is defined for such pairs of operators and it
equals the topological index of the pair of their symbols.
