On linear combinations of two idempotent matrices over an arbitrary field

Given an arbitrary field K and non-zero scalars α and β, we give necessary and sufficient conditions for a matrix A∈Mn(K) to be a linear combination of two idempotents with coefficients α and β. This extends results previously obtained by Hartwig and Putcha in two ways: the field K considered here is arbitrary (possibly of characteristic 2), and the case α≠±β is taken into account.