Abstract: Let Bm×n  be the set of all m×n Boolean matrices;R(A) denote the row space of A∈Bn,|R(A)| denote the cardinality of R(A),m,n  be positive integers,and k be non negative integers.In this paper,we prove the following three results:(1) let A∈Bn×n,m,(ⅰ) if A  is the idempotent matrix,i.e.,A2=A,then |R(Am)|=|R(A)|;(ⅱ) if A  is the involutory  matrix,i.e.,A2=I,then |R(Am)|=|R(A)|  when m is an odd number or |R(A)|=2n when m is an even number;(2) let  A∈Bm×nbe k of  the numbers  of  1,0≤k≤min{m,n},and each row and column is at most one of the numbers of 1 in A,then |R(A)|=2k; (3) let A∈Bn×n be the partitioned matrix as A=(O OO A1),A1=(aij)k×k,aij=0(i>j),aij=1(i≤j),i,j=1,2,…,k,then |R(A)|=k+1.

Key words: Boolean matrix, row space, cardinality of a row space, permutation matrix