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)

⇒ AI_n = O   ($\because$ BB^–1 = I_n) 

⇒ A = O

But A is a non-null matrix. Hence, B is a singular matrix. Similarly it can be shown that A is a singular matrix.