Identities and Inequalities on the Rank for some Submatrices of a Matrix over a Division Ring

Abstract:

      This paper studies the "substructure" of a matrix over the division ring, and provides an inequality on the rank for the submatrix of the product of two matrices, and an identity on the rank for the submatrix of an inverting matrix. An inequality On the rank of the submatrix of a nonsingular matrix is also given. Some applications of the above conclusions is provided also, one of the interesting results is that the rank of every (n- 1 ) × (n- 1) submatrix of an n × n nonsingular matrix is at least n-2.