Let $n \geq 2$ be an integer. Given an integer $k$ there exists an $n \times n$ matrix $A$ with integer entries such that $\operatorname{det} A = k$ and the first row of $A$ is $(1, 2, \ldots, n)$.
Let $n \geq 2$ be an integer. Given an integer $k$ there exists an $n \times n$ matrix $A$ with integer entries such that $\operatorname{det} A = k$ and the first row of $A$ is $(1, 2, \ldots, n)$.