  The main result of the paper proves that the monoid of all 2 x 2 tropical
matrices satisfies a non-trivial semigroup identity, unlike the case of
matrices over infinite fields. We exploit a strong connection between the
monoid of all 2 x 2 tropical matrices and the bicyclic monoid to show that a
variant of Adian's identity for the bicyclic monoid holds in our case.We use
generic matrices of tropical polynomials to prove our result.
