Ranking of matrix associated to non homogenous linear equations

Question

Consider a linear system like $Ax=b$. Matrix $A$ is a non trivial square matrix of size $n$.

If we know that there are $n-1$ linear independent solutions of the system above, what can we say about the rank of matrix $A$?

I know that the rank of $A$ is at most $2$. I wonder if we conclude that rank of $A$ is 1 or not.

Thanks to the comment, now I add one condition that $n>2$.

Thanks in advance.

Answer 1

No. Consider, $n=2$ and $v=(1,1)$, $A=diag(v)$ and $b=v$.

Then the only one solution is $x=v$ and the rank of $A$ is $2$.

If $n>2$, consider $v=(1,1,0,...,0)$, $A,b$ as before. Consider $e_i$ the unit basis vectors. Then $v, v+e_i$ for $i>2$ form $n-1$ independent solutions. But the rank is still $2$ :)