We study properties of the Drazin index of regular elements in a ring with a unity 1. We give expressions for generalized inverses of 1 − ba in terms of generalized inverses of 1 − ab. In our development we prove that the Drazin index of 1 − ba is equal to the Drazin index of 1 − ab.2000 Mathematics subject classification: primary 15A09; secondary 16U99.

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An n × n matrix is called an N 0-matrix if all its principal minors are nonpositive. In this paper, we are interested in N 0-matrix completion problems, that is, when a partial N 0-matrix has an N 0-matrix completion. In general, a combinatorially or non-combinatorially symmetric partial N 0-matrix does not have an N 0-matrix completion. Here, we prove that a combinatorially symmetric partial N 0-matrix, with no null main diagonal entries, has an N 0-matrix completion if the graph of its specified entries is a 1-chordal graph or a cycle. We also analyze the mentioned problem when the partial matrix has some null main diagonal entries.

An n × n real matrix is said to be totally nonpositive if every minor is nonpositive. In this paper, we are interested in totally nonpositive completion problems, that is, does a partial totally nonpositive matrix have a totally nonpositive matrix completion? This problem has, in general, a negative answer. Therefore, we analyze the question: for which labeled graphs G does every partial totally nonpositive matrix, whose associated graph is G, have a totally nonpositive completion? Here we study the mentioned problem when G is a chordal graph or an undirected cycle.