Suppose the last column of $AB$ is entirely zero but $B$ itself
has no column of zeros. What can you say about the columns of $A$?
Correct answer: The columns of $A$ are linearly dependent.
I don't see the correlation between those two. If it's linearly dependent, there exists a solution where the columns multiplied by a scalar equals 0, but I don't see how a column of all 0s here shows that it's linearly dependent.
Best Answer
Hint: Write out the definition of the matrix product, then try to rewrite the last column of the product as a linear combination of the columns of $A$. You'll find the non-trivial linear combination that results in 0.