  In some matrix formations, factorizations and transformations, we need
special matrices with some properties and we wish that such matrices should be
easily and simply generated and of integers. In this paper, we propose a
zero-sum rule for the recurrence relations to construct integer triangles as
triangular matrices with involutory, idempotent, nilpotent and unipotent
properties, especially nilpotent and unipotent matrices of index 2. With the
zero-sum rule we also give the conditions for the special matrices and the
generic methods for the generation of those special matrices. The generated
integer triangles are mostly newly discovered, and more combinatorial
identities can be found with them.
