Given $AB=0$. Prove that every column of $B$ is a solution for $Ax=0$ I am studying for my test by going through some old tests from my university and in one of the tests there's the following question:

Given $AB=0$ prove that every column of $B$ is a solution for $Ax=0$

And I am very confused, isn't the solution to this problem is obvious? what are they expecting as an answer...?
Isn't it simply:
$AB=0 \rightarrow A[B]^c_i=[0]^c_i$ which shows that every column of $B$ is a solution for $Ax=0$

Here is a formal proof.
Let $e_j$be the column vector with all entries zero, except for the $j$th one, which is $1$.
Then $AB=0$ implies $0=0e_j=(AB)e_i=A(Be_j)=AB_j$, where $B_j$ is the $j$th column of $B$.
The proof relies on the associativity of matrix multiplication.
