Let $v$ be a vector and $A$ be a real matrix generated with some algorithm from components of the vector i.e. $A=f(v)$.
Now I'm interested in such matrices $A$ that for them
$\det(A)=\det(f(v))= \det(f(v+k\mathbf{1}))$ for any real $k$,
where $\mathbf{1}$ is the vector consisted of only $1$ so the $v+k\mathbf{1}$ is the vector $v$ translated by the vector $k\mathbf{1}$.
Question:
what are examples of matrices with such property ?
I've collected some:
the first matrix would be the matrix $2\times{2}$ constructed from a $4\times{1}$ vector with some specific properties from this MSE question,
the second matrix would be $n\times{n}$ Vandermonde matrix constructed from a general form of $n\times{1}$ vector,
- the third matrix would be $n\times{n}$ matrix from this MSE question constructed from a vector as above.
What other non-trivial examples of such matrices can be presented?
