Let R be a principal ideal ring, A a symmetric t-by-t matrix over R, B a t-by-(n — t) matrix over R such that the t-by-n matrix (A, B) is primitive. Newman  proved that (A, B) may be completed (as the first t rows) to a symmetric n-by-n matrix of determinant 1, provided that 1 ≦ t ≦ n/3. He showed that the result is false, in general, if t = n/2, and he asked to determine all values of t such that 1 ≦ t ≦ n and the result holds. It is shown here that these values are exactly t satisfying 1 ≦ t < n/2. Moreover, the result is proved for a larger (than the principal ideal rings) class of commutative rings, namely, for the rings satisfying the second stable range condition of Bass . Also, it is observed that Theorems 2 and 3 of [2, p. 40] proved there for principal ideal rings are true for this larger class of rings, as well as the basic result of [2, p. 39].
All Science Journal Classification (ASJC) codes
- Applied Mathematics