This study theoretically investigates Fano resonances and dips of an Au-SiO2-Au nanomatryoshka that is excited by a nearby electric dipole. An analytical solution of dyadic Green's functions is used to analyze the radiative and nonradiative power spectra of a radial dipole in the proximity of a nanomatryoshka. From these spectra, the plasmon modes and Fano resonances that accompany the Fano dips are identified. In addition, the scattering and absorption spectra of a nanomatryoshka that is illuminated by a plane wave are investigated to confirm these modes and Fano dips. Our results reveal that a Fano dip splits each of the dipole and quadrupole modes into bonding and anti-bonding modes. The Fano dip and resonance result from the destructive interference of the plasmon modes of the Au shell and the Au core. The Fano factors that are obtained from the nonradiative power spectra of the Au shell and the Au core of a nanomatryoshka are in accordance with those obtained from the absorption cross section spectra. Moreover, these Fano factors increase as the plasmonic coupling of the Au shell with the core increases for both dipole and quadrupole modes.