[Note] the below question is Multiple choice question, that is it may have more than one correct options.
Question: let $A$ be a $3×4$ matrix with integer entries and $b$ is a $3×1$ matrix with integer entries. Suppose that the system $Ax=b$ has complex solution then, which of following are true
(a) $Ax =b$ has an integer solution
(b) $Ax =b$ has rational solution.
(c) the set of real solution to $Ax=0$ has basis consisting of rational solution.
(d) if $b≠0$ then $A$ has positive rank.
The above question was asked in "Csir- Net Mathematics examination (Dec-2014) India"
I just know that, system $Ax = b$ has solution if $rank(A) = rank(A|b)$. Please help me. I am unable to solve it.
(a) is false. Suppose that the entries of $b$ are all equal to $1$ and that the entries of $A$ are even numbers. Then no integer solution can exist.
(b) and (c) is true. Just use row reduction (and the fact that the entries of $A$ are integers) to prove it.
(d) is also true. This is so because if $Ax=b$ has a solution, then $A$ cannot be the null matrix (because $b\neq0$) and therefore its rank is at least $1$.