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

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 $1 \leq t \leq n/3$. He showed that the result is false, in general, if $t = n/2$, and he asked to determine all values of $t$ such that $1 \leq t \leq n$ and the result holds. It is shown here that these values are exactly $t$ satisfying $1 \leq t \leq n/2$. 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].