A dual frequency and dual directional antenna is an antenna with maximum gain in two different directions for two different frequencies. Maximum gain will be reached by the matrix method. This will be applied for two different situations. In the first we maximize the gain at one frequency and one direction under the constraint that the gain at the other frequency and direction has a given value. In the second we maximize the gain at one frequency and one direction under the constraint that it has a given relation with the gain in the other. A comparison between the two methods shows some practical considerations.