  In contrast to the situation in classical linear algebra, not every
tropically non-singular matrix can be factored into a product of tropical
elementary matrices. We do prove the factorizability of any tropically
non-singular 2x2 matrix and, relating to the existing Bruhat decomposition,
determine which 3x3 matrices are factorizable. Nevertheless, there is a closure
operation, obtained by means of the tropical adjoint, which is always
factorizable, generalizing the decomposition of the closure operation * of a
matrix.
