Here we present the rigorous electrodynamical solution of diffraction problem about the microwave scattering by a multilayered cylinder. The number and thickness of layers is not limited. We offer the solution when the central core of multilayered cylinder can be made of different isotropic materials as a metamaterial, a ceramic matter or a semiconductor as well as of a perfect metal. The isotropic coated layers can be of strongly lossy materials. The signs of the complex permittivity and the complex permeability can be negative or positive in different combinations. Here we present dependencies of the scattered power of the incident perpendicularly and parallel polarized microwaves by the metamaterial-glass cylinder on signs of metamaterial permittivity as well as permeability. Here are also presented the glass layer absorbed power and the metamaterial core absorbed power dependent on the hypothetic metamaterial permittivity and permeability signs at the wide range frequencies 1-120 GHz. The metamaterial core of cylinder has a radius equal to 0.0018 m and the thickness of the coated acrylic-glass layer is 0.0002 m. We have found some conditions when the scattered-power has minimal values and the absorbed power by the coated acrylic glass layer is constant in a very wide frequency range. We have discovered that the glass layer absorbed power decreases with increasing of the frequency at the range 1-120 GHz for both microwave polarizations.
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