Matrix Multiplication Non-commutativity

Matrix multiplication is associative and distributive but not commutative, a fundamental distinction from scalar arithmetic emphasized in JEE Advanced. For general square matrices A and B, the product AB depends on the order because rows of the first multiply columns of the second, and reversing the order changes which combinations are formed. Hence AB need not equal BA, and explicit examples readily show inequality. Option AB always equals BA is false, since commutativity fails for most matrix pairs. Option AB = 0 implies A = 0 or B = 0 is false, because non-zero matrices can multiply to the zero matrix, unlike real numbers. Option (AB)^T = A^T B^T is false; the correct reversal law is (AB)^T = B^T A^T. Hence the always-true statement is that AB need not equal BA. Plausibility check: taking A and B as two distinct rotation-like or nilpotent matrices produces AB ≠ BA, and the existence of zero-divisor matrices confirms the other options fail, validating the chosen statement. Non-commutativity is the single most important way matrix algebra departs from ordinary arithmetic, and it underlies why solving matrix equations requires care about left versus right multiplication. The existence of zero divisors, where a product of non-zero matrices is the zero matrix, further breaks intuition carried over from real numbers and is a recurring source of conceptual JEE Advanced questions.