Consider a system of linear equations $Ax=b$ where $A$ is an $n\times n$ matrix. Suppose that $b$ is a non-zero vector such that $A^t b=0$. Which is true about any such system?
I am given 5 choices for what this means: The system has infinitely many solutions, is inconsistent, is consistent, is over determined, or is in row echelon form.
Unfortunately, I have yet to completely rule out a single one. I don't think it can be over determined, because then wouldn't the equations be undefined based on the fact that if there were more equations than unknowns then $Ax$ would not be possible. I don't recall any theorems discussing whether the system is consistent or inconsistent using the transposition and b, which stems to the infinitely many solutions.