We consider the stability problem for the difference system x n = A x n − 1 + B x n − k , where A , B are real matrixes and the delay k is a positive integer. In the case A = − I , the equation is asymptotically stable if and only if all eigenvalues of the matrix B lie inside a special stability oval in the complex plane. If k is odd, then the oval is in the right half-plane, otherwise, in the left half-plane. If ‖ A ‖ + ‖ B ‖ < 1 , then the equation is asymptotically stable. We derive explicit sufficient stability conditions for A ≃ I and A ≃ − I .