$n$ by $n$ matrices such that $Ax=0$ implies $Bx=0$, then what can we say about $A$ and $B$? Ok, this is one of the interesting questions I encountered in my GRE Maths exam 2 weeks ago. Fix an $x\in \mathbb{R}^n$ (doesn't have to be $0$), and $A$, $B$ are $n$ by $n$ matrices such that $Ax=0$ implies $Bx=0$, then what can we say about $A$ and $B$? I think the choices were: $\exists C$ an $n$ by $n$ matrices such that: 
a) $A=BC$, b) $A=CB$, c) $B=CA$, d) $B=AC$, e) $B=C^{-1}AC$.
Well, the set of matrices $D$ such that $Dx=0$ form an left ideal, but I only know what bi-ideals look like in this case. Also the wording the this question is funny, we dont know if $A$ is in the ideal or not, we just know if $A$ is in the ideal then $B$ is in the ideal.
Edit: As we discussed below, I think I misunderstood the question in the exam (although I don't have the actually paper now). $x$ is probably not fixed.

Correct answer: (c)
$Ax=0$ is equivalent to "$x$ is perpendicular to all the rows of $A$". Hence, 
the statement
$$
\forall x\in\mathbb R^n (Ax=0\Longrightarrow Bx=0)
$$
tells us that the space spanned by the rows of $B$ is a subspace of the space spanned by the rows of $A$. Hence, each row $b_i$, $i=1,\ldots,n$, the of $B$ is a linear combination of the rows of $A$. Thus, $b_i=c_iA$, where $c_i$ is a vector of the same dimension.
Therefore, $B=CA$, where $C$ is the matrix with rows $c_1,\ldots,c_n$. 
