transpose of System of equations

Question

Let $A$ be an $m\times n$ matrix of rank $n$ with real entries. Choose the correct statements. 1. $Ax=b$ has a solution for any $b$. 2. $Ax=0$ does not have a solution. 3. If $Ax=b$ has a solution, then it is unique. 4. $y'A=0$ for some non zero $y$, where $y'$ denotes the transpose vector of $y$.

Option 1 and 2 are false for sure. Option 3 is correct. Any one kindly explain option 4.

Answer 1

Option 4 could be false. Take $m = n = 2$ and let $A$ be the $2 \times 2$ identity matrix. Then $\mathbf y' = \mathbf 0'$ so that $\mathbf y = \mathbf 0$.

Answer 2

If you transpose both sides you get: $$ y' A = 0 \iff (y' A)' = 0' \iff 0 = A' y'' = A' y $$ so 4. will not hold in general. ($m = n$, $A'$ invertible)