This study considers an optimal output anti-synchronisation problem for linear multi-agent systems with non-identical dynamics and antagonistic interactions. It is assumed that the agents are non-introspective, i.e. they cannot get their own states or outputs, and can only get a linear combination of relative outputs in the neighbourhood. First, a state transformation is made, which aims at reducing the dimension of the model underlying the output anti-synchronisation errors by removing redundant modes that have no effect on the output anti-synchronisation errors. Then, optimal state feedback controllers are proposed for agents to realise the output anti-synchronisation by constructing discounted performance functions and by solving the discounted algebraic Riccati equation. Moreover, by using an iterative technique, an offline algorithm is provided to solve the linear quadratic tracking problem. In addition, a distributed node-based adaptive observer is proposed to produce an estimate of the state with only relative output information of neighbouring agents by utilising a high-gain approach. The convergence of the proposed adaptive observer is analysed with the help of a Lyapunov function method. Finally, numerical simulations are carried out to demonstrate the effectiveness of the proposed approach.