Let A  and  B  be two non-zero square matrices. If the productAB is a null matrix, then

Let B  be non-singular, then B−1 exists. Now, AB=O  (given)⇒         (AB)B−1=OB−1                        (post-multiplying both sides by B−1 )⇒    A(BB−1)=O       (by  associativity)⇒    AIn=O                  (∵  BB−1=In) ⇒   A=O But A is non-null matrix. Hence, b is a singular matrix.Similarly, it can be shown that A is a singular matrix.