Say you have an linear operator $\mathcal{A}: \mathbb{R}^{m \times n} \rightarrow \mathbb{R}^p$ and a vector $\vec b \in \mathbb{R}^p$, with $p < mn$. Assume that the linear equation $$\mathcal{A}(M) = \vec b$$ has at least one solution matrix $M_*$ with rank $r$.
My question is: Is it possible that there is only one unique solution matrix $M_{X}$ with rank $r+1$ ? My intuition screams "NO!", however I can't manage to prove it.
Any ideas?
If the solution set is the affine space $$ \begin{pmatrix} x & 1 & 1\\ 1 & x & 1\\ 1 & 1 & x \end{pmatrix} $$ then it has rank $1$ for $x=1$, but rank $2$ only for $x=-2$.