The difference equations \(\phi_{n+1}=L_ n\phi_ n,\) \(\phi_{nt}=N_ n\phi_ n\) and \(L_{nt}=N_{n+1}L_ n-L_ nN_ n\) are considered. The so called Darboux transformation, represented by a nonsingular matrix \(M_ n\) and defined by certain algebraic conditions, is applied to the original system of equations. It is shown that the considered system is invariant in its construction and form under Darboux transformation. Several examples are included.