We call the matrix A as(m,l)rank-idempotent matrix if there exist natural numbers m,l(m>l)such that r(A~m)=r(A~l);when A~m=A~l,then A is called(m,l)idempotent matrix.

We call the matrix A as(m,l)rank-idempotent matrix if there exist natural numbers m,l(m>l)such that r(A~m)=r(A~l);when A~m=A~l,then A is called(m,l)idempotent matrix.

Applying the method of the elementary transformation of block matrix,we derive an identity of the rank about the power of matrix,thereby obtain the necessary and sufficient condition of m-idempotent matrix and m-involutory matrix,furthermore,generalize and improve the corresponding results.

Let A be an idempotent matrix,B be an l-potent matrix(i.

The nonsingularity of linear combinations of two idempotent matrices is discussed by using null space of matrices and thus proves the necessary and sufficient conditions of nonsingularity of linear combinations of two idempotent matrices.

This paper shows several rank egualities for idempotent and involutory matrices and gives several mecessary and sufficient conditions for aP+bQ bing invertible (P and Q are idempotent matrices);and gives several necessary and sufficient conditions for A+B+2I_n being invertible(A and B are involutory matrices).

In this paper we discuss the properties of idempotent matrices, EP matrices, GP matrices,HGP matrices,block matrices and nilpotent matrices over skew field.