Q: If $A$ is a $3 \times 3$ matrix and $y$ is a vector in $\mathbb{R}^3$ such that $Ax = y$ does not have a solution, then there exists no vector $z$ in $\mathbb{R}^3$ such that the equation $Ax = z$ has a unique solution.
This is true right? If $A$ doesn't span $\mathbb{R}^3$ then the only possibility is either no solution or infinite solutions on an $\mathbb{R}^2$ plane?