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 prove the factorizability of any tropically non-singular 2\times 2 matrix, relating to the tropicalization of the existing Bruhat decomposition, and determine which 3\times 3 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 tropical closure operation: Kleene star.