Suppose A(t) is T-periodic. Prove that for every t0 and every T-periodic function f(t), there exists x0 such that the solution of x(t) = A(t)x(t) + f(t), x(t0) = x0 is T-periodic if and only if f(t) is such that for all T-periodic solution z(t) of the adjoint state equation z(t) = -AT(t)z(t), z(t0) = z0.