I have an equation of the form:
$$A(s,t)x = b(s)$$
where $A$ is a real matrix, $x, b$ vectors and $s, t$ scalar parameters.
While generally speaking, if $A$ is singular, the equation generically has no solution, for my specific $A$ and $b$ I observe that $$\det(A(s,t))=0 \quad \Rightarrow\quad \exists x \ |\ A(s,t)x=b(s)$$ or, in other words, whenever $(s,t)$ makes $A$ singular, there always exists a solution $x$ to the equation. (Of course when $A$ is not singular there is also a solution: $x=A^{-1}b$.)
The question is, how to characterize the relationship between $A$ and $b$? $b$ is always in the image of $A$, but can I write mathematically in a more explicit way?