Let $A$ be a $3\times 4$ and $b$ be a $3\times 1$ matrix with integer entries.Suppose that the system $Ax=b$ has a complex solution. Then which of the following are true? (CSIR December 2014)
- $Ax=b$ has an integer solution.
- $Ax=b$ has a rational solution.
- The set of real solutions to $Ax=0$ has a basis consisting of rational solutions.
- If $b$ is not equal to zero then $A$ has positive rank.
Is it possible to say that 1 and 2 are true as $Ax=b$ has a complex solution? If not what we have to understand from the statement "the system $Ax=b$ has a complex solution " ? What about 3 and 4 ?