The solution to your problem is as follows -
Let A and B be two non-empty (or non-null) matrices.
By the hypothesis of the question, AB=0
Now, Assume that A is non-singular.
=> A-1 exists.
Therefore, multiply both sides of equation by A-1
But this is a contradiction to the fact that B is non-empty matrix. Hence A can't be non-singular matrix.
Similarly, assume B to be non singular.
=> B-1 exists.
Multiply both sides by B-1,
Again, this is contradiction to the fact that A is non-empty. Therefore, B can't be non-singular.