Results On(a,b)-matrices And Special Invertible Matrices

In this thesis, we study the following three problems:(1) the number of b in an (a, b)-matrix with a given rank;(2) the form of an invertible matrix A=(aij)∈Mn satisfying A-1=(a-1ij)(We assume that0-1=0);(3) the form of an integer matrix A satisfying Ak=I where k is a certain integer and I is the identity matrix. We obtain partial results for these problems in our thesis.The rank of a matrix does not change if we multiply or divide the matrix by a nonzero number. Therefore, it suffices to consider the following three cases:(0,1)-matices,(-1,1)-matrices and (a,1)-matrices (-1<a<1and a≠0). For problem (1), we prove that the set of possible numbers of1of an (a,1)-matrix (a≠0) with order n and rank k (2≤k≤n) is the integer interval [k-1, n2-k+1] and we consider the special case of rank1; we also show that the set of possible numbers of1of a nonsingular symmetric (a,1)-matrix (a≠0) of order n (n≥4) is the integer interval [n-1, n2-n+1] and we consider the special case of rank3; in addition, we give possible numbers of1of singular symmetric (-1,1)-matrices of order n and rank1and3respectively. For problem (2), we give a necessary and sufficient condition when A is symmetric; we also characterize all the possible matrices when the rank is2and3respectively; we, in addition, give several kinds of matrices which meet or do not meet the condition. For problem (3), we obtain a necessary and sufficient condition for Ak=I when A is a (0,1)-matrix; then we give a necessary and sufficient condition for A2=I when A is a symmetric integer ma-trix; furthermore, we utilize Matlab to generate some matrices which meet the condition.