In the following paper the circular representation of basic positive and basic negative alternating cycle matrices is determined. This representation enables the calculation of the multiplication tables of basic positive and basic negative alternating cycle matrices. Surprisingly, the associated matrices of these multiplication tables are alternating cycle matrices themselves. Several further properties are derived. The multiplication of positive and negative alternating cycle matrices follows the rule of signs of the multiplication of positive and negative numbers. The multiplication of positive alternating cycle matrices is commutative. The multiplication of negative alternating cycle matrices is symmetrical.