Copyright © 2009 Marina Arav et al. This is an open access article distributed under the
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Let be an real matrix and let be the set of column indices of the zero entries of row of . Then the conditions for all are called the (row) Zero Position Conditions (ZPCs). If satisfies the ZPC, then is said to be a (row) ZPC matrix. If satisfies the ZPC, then is said to be a column ZPC matrix. The real matrix is said to have a zero cycle if has a sequence of at least four zero entries of the form in which the consecutive entries alternatively share the same row or column index (but not both), and the last entry has one common index with the first entry. Several connections between the ZPC and the nonexistence of zero cycles are established. In particular, it is proved that a matrix has no zero cycle if and only if there are permutation matrices and such that
PHQ is a row ZPC matrix and a column ZPC matrix.