### Abstract

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 [2] 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 [1]. 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].

Original language | English (US) |
---|---|

Pages (from-to) | 189-196 |

Number of pages | 8 |

Journal | Proceedings of the American Mathematical Society |

Volume | 97 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1986 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

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*Proceedings of the American Mathematical Society*, vol. 97, no. 2, pp. 189-196. https://doi.org/10.1090/S0002-9939-1986-0835863-1

**An answer to a question of M. Newman on matrix completion.** / Vaserstein, L. N.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An answer to a question of M. Newman on matrix completion

AU - Vaserstein, L. N.

PY - 1986/6

Y1 - 1986/6

N2 - 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 [2] 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 [1]. 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].

AB - 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 [2] 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 [1]. 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].

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U2 - 10.1090/S0002-9939-1986-0835863-1

DO - 10.1090/S0002-9939-1986-0835863-1

M3 - Article

AN - SCOPUS:84968482104

VL - 97

SP - 189

EP - 196

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 2

ER -