Given a $k\times n$ integer primitive matrix $\bf{A}$ (i.e., a matrix can be extended to an $n\times n$ unimodular matrix over the integers) with the maximal absolute value of entries $\|\bf{A}\|$ bounded by {an integer} $\lambda$ from above, we study the probability that the $m\times n$ matrix extended from $\bf{A}$ by appending other $m-k$ row vectors of dimension $n$ with entries chosen randomly and independently from the uniform distribution over $\{0, 1,\ldots, \lambda-1\}$ is still primitive. We present a complete and rigorous proof of a lower bound on the probability, which is at least a constant for fixed $m$ in the range $[k+1, n-4]$. As an application, we prove that there exists a fast Las Vegas algorithm that completes a $k\times n$ primitive matrix $\bf{A}$ to an $n\times n$ unimodular matrix within expected $\tilde{O}(n^{\omega}\log \|\bf{A}\|)$ bit operations, where $\tilde{O}$ is big-$O$ but without log factors, $\omega$ is the exponent on the arithmetic operations of matrix multiplication.

翻译：给定一个$k\times n$整数原始矩阵$\bf{A}$（即可以扩展为整数域上$n\times n$幺模矩阵的矩阵），其中$\|\bf{A}\|$表示矩阵元素最大绝对值，且被整数$\lambda$从上方界定时，我们研究将该矩阵通过附加$m-k$个维度为$n$的行向量（这些行向量元素从$\{0,1,\ldots,\lambda-1\}$均匀分布中独立随机选取）扩展为$m\times n$矩阵后，该矩阵仍为原始矩阵的概率。我们对概率的下界给出了完整且严谨的证明，该下界在$m$属于区间$[k+1,n-4]$时为常数。作为应用，我们证明存在一种快速拉斯维加斯算法，能在期望$\tilde{O}(n^{\omega}\log \|\bf{A}\|)$比特操作内将$k\times n$原始矩阵$\bf{A}$补全为$n\times n$幺模矩阵，其中$\tilde{O}$表示忽略对数因子的大$O$记号，$\omega$为矩阵乘法算术运算的指数。