We know,

(A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A)

Here A and B have an element in common i.e., n(A ∩ B) = 1 = (B ∩ A)

So, n((A × B) ∩ (B × A)) = n((A ∩ B) × (B ∩ A)) = n(A ∩ B) × n(B ∩ A) = 1 × 1 = 1

That means, A × B and B × A have an element in common if and only if A and B have an element in common. [Proved]