Fluorescence molecular tomography (FMT) is a hot research topic in molecular imaging applied to precision medicine. Due to the seriously ill posed FMT inverse problem, background fluorescence noise often degrades FMT reconstruction results greatly. After analyzing FMT reconstruction methods based on the finite element method, we propose a method for suppressing background fluorescence using low rank matrix completion technology. The proposed method builds an incomplete boundary observation matrix which columns correspond to different excitation sources. Then, a low rank matrix completion algorithm is employed to complete the matrix and suppress the background fluorescence. At last, a new FMT inverse problem is formed by the denoised boundary observation matrix, with which the fluorescent targets are reconstructed. The new FMT inverse problem is applied to reconstruct single and double fluorescent targets, numerical experiments illustrate that the reconstruction results are improved greatly.