On the idempotency, involution and nilpotency of a linear combination of two matrices.

This paper deals with the idempotency, involution and nilpotency of two matrices. In this paper, by introducing a new idea into the classical block technique, we characterize all situations in which a linear combination of the formis(Q1) square-zero when P and Q are both idempotent;(Q2) square-zero when P is tripotent and Q is idempotent, respectively;(Q3) idempotent when P is tripotent and Q is involutory, respectively;(Q4) involutory when P is tripotent and Q is involutory, respectively.In addition, we point out several other related problems that the reformed method in this paper applies. [ABSTRACT FROM AUTHOR]