On the probability of generating a primitive matrix

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.