You could use some facts about rank, namely that the rank of a matrix is the maximum number of independent columns and rank AB<= rank B.
This might be overkill because depending on which definition of rank you use, it could be hard to prove either one of these. Sometimes rank is defined as the largest number of independent columns, but then I don't think showing that rank AB<=rank B is too easy. If rank is defined as the size of a largest minor with nonzero determinant, then rank AB<=rank B follows quickly but the first fact might not. Maybe someone can come up with a simpler solution.