The nonlocal Euler-Bernoulli beam model is used to study the vibration instability characteristics of fluid transporting fixed-fixed supported single-layer carbon nanotubes (SWCNTs) embedded in elastic matrix under longitudinal magnetic field. Considering the magnetic field force and small-scale effect, the elastic matrix is equivalent to Pasternak's property model, and the higher-order vibration differential equation and boundary condition of the fluid-structure coupling system are established by using Hamilton principle. The differential transformation method (DTM) is used to solve the equation, and the stability of SWCNTs under the coupling of elastic matrix, longitudinal magnetic field and small-scale effect is studied. The numerical results indicate that the magnetic field and the elastic matrix improve the stability of the system. The increase of small-scale coefficient will reduce the stability of the system to a certain extent. Further research shows that the influence of longitudinal magnetic field on the stability of flow transporting SWCNTs embedded in elastic matrix will be restrained to varying degrees with the enhancement of the elastic matrix. For the small-scale coefficient, the two parameters of the elastic foundation have the opposite effect, that is, the shear parameter suppresses the influence of small-scale on the stability of the fluid-conveyed carbon nanotube system, while the elastic parameter amplifies the influence.