In this paper, we study the existence and uniqueness of signed solutions for systems of ω-linear balances over the symmetrized omega algebra. Furthermore, we generalize the main algebraic features of balances and define the rank, linear combination, linear dependence and linear independence over the symmetrized omega algebra. Definitions and some related properties of the determinant and adjoint matrix over the symmetrized omega algebra are introduced.

tropical geometry, idempotent semirings, omega algebra, symmetrized omega algebra, ω-linear dependence, ω-linear independence, ω-linear balances, matrix.