Let A and B be two non-null square matrices. If the product AB 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 = OBut A is a non-null matrix. Hence, B is a singular matrix.Similarly it can be shown that A is a singular matrix.