Anticommutative implication

Chef is impressed by mathematical logic and linear algebra. He wants to combine his favorite subjects, so he introduced the concept of logical operations on boolean matrices. He defines A → B to be a matrix C such that Cij = Aij → Bij and ¬ A to be a matrix B such that Bij = ¬ Aij Now Chef wants to study such system of equations: A → X = ¬ O and X → A = O Where O is matrix such that all its ent