Given a vector $a_1=(k_1,\ldots,k_n)^T$ of coprime integers. Are there $a_2,\ldots,a_n \in \mathbb{Z}^n$ such that that the matrix $A := (a_1,\ldots,a_n) \in \mathbb{Z}^{n \times n}$ is regular, i.e. $A \in GL_n(\mathbb{Z})$ ?
In case $n=2$ this is true because there are integers $l_1,l_2$ s.t. $k_1l_1 + k_2l_2 = 1$. Then $a_2=(-l_2 , l_1)^T$ will do.