Claude Berge defines a (0,1) matrix A to be linear if A does not contain a 2×2 submatrix of all ones. A(0,1) matrix A is balanced if A does not contain a square submatrix of odd order with two ones per row and column. The contraction of a row i of a matrix consists of the removal of row i and all the columns that have a 1 in the entry corresponding to row i. In this paper the authors show that if a linear balanced matrix A does not belong to a subclass of totally unimodular matrices, then A or AT contains a row i such that the submatrix obtained by contracting row i has a block-diagonal structure.
