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We have studied the Compton scattering ($\gamma e^- \longrightarrow \gammae^-$) at the tree level in a homogeneous background of strong magnetic field($|eB| \gg { m }^{ 2 },~\text{m is the mass of electron}$) through the S-matrixapproach. For that purpose, using the Schwinger propagator for the electron, wehave first calculated the square of the S-matrix element in the Landau gauge bysumming over the final states of electron and photon and averaging over theinitial states of the same. In the strong magnetic field, only the lowestLandau level for electron is considered. Finally we have computed thecrosssection for Compton scattering as a function of initial photon energy forthe different strengths of strong magnetic fields, where we have found that thecrosssection in vacuum gets decreased due to the presence of strong magneticfield. However, for a fixed initial photon energy, the crosssection increaseslinearly with the magnetic field.

S-Matrix approach to Compton scattering at the tree level in a strong magnetic field