On the spectrum and pseudoinverse of a special bordered matrix

The characteristic polynomial of a low rank bordered matrix of special structure is determined by elementary means. This result includes as a special case formula (13) in [A. Farkas, P. Rózsa, Data perturbations of matrices of pairwise comparisons, preprint, 2000] obtained there by the Sherman–Morrison formula. Another application is a simple formula for the pseudoinverse of these bordered matrices.