$A y= b$ in $C(X)$

Question

Let $X$ be a compact Hausdorff topological space, and $C(X)$ denote the ring of all complex valued continuous functions on $X$. If $A\in C(X)^{m\times n}$, $b\in C(X)^{m\times 1}$, and for all $x\in X$, $b(x)$ belongs to the range of $A(x)$, then does there exist a $y\in C(X)^{n \times 1}$ such that $Ay =b$?

Answer 1

If $X=[0,1]$, $b(x)=\sqrt{x}$, $A(x)=x$, then there is no solution.

(Thanks again Daniel Fischer!)