Number of Negative Entries in A²≤0

Let A be a real matrix or a sign pattern of order n. N_-（A） denotes the number of negative entries in A. In 1972 R DeMarr and A Steger conjectured: If A is a real matrix of order n such that A²≤0, then(N_-（A²）≤）（n-1）²+1. Now the conjecture is proved to be true when A is reducible or a matrix of order n≤3 and some sufficient conditions for N_-（A²）≤（n-1）²+1 are given. It is also proved that N_-（A²）≤n²-（4n+）5 when A is a reducible combinatorially symmetric sign pattern such that A²≤0, and the extreme sign patterns are characterized.