Final answer:
Matrix A and B commute, meaning AB = BA, by definition. The statement 'AB = BA if and only if A and B commute' is true by definition of commuting matrices.
Step-by-step explanation:
To prove that for square matrices A and B, AB = BA if and only if A and B commute, we need to understand matrix multiplication and the properties of commutativity. By definition, two matrices A and B commute when their product is the same regardless of the order of multiplication, meaning AB=BA. Commuting is not a common property in matrix multiplication, as most matrices do not commute.
The phrase 'AB = BA if and only if A and B commute' is mathematically redundant because it essentially says 'AB = BA if and only if AB = BA'. The assertion is trivially true because it's just restating the definition of commuting matrices.