What can be said about a matrix with a constant column of ones with entries from a finite field?

Question

I am working with matrices of the following structure:

$A = \begin{pmatrix} 1&\alpha_{21}&\cdots&\alpha_{n1}\\ 1&\alpha_{22}&\cdots&\alpha_{n2}\\ \cdots&\cdots&\ddots&\vdots\\ 1&\alpha_{n2}&\cdots&\alpha_{nn} \end{pmatrix}$

where the $\alpha_{ij}$ come from a finite field $\mathbb{F}_q$ with $q$ a prime or prime power, and the first column is all 1. What can be said about this kind of matrix? Does this class of matrices have a name?

I am mostly interested in determining when $A$ is singular, but other properties would be useful too.

Answer 1

Nothing special about such a matrix. If a matrix has a single nonzero entry, then you can put it in this form with elementary row operations and a column permutation, which does not have an effect on whether or not it is singular.