Matrix A is 3×4 and matrix B is 4×2. Which of the following is correct?
ABoth AB and BA are defined, and both produce 3×2 matrices
BAB is defined and produces a 3×2 matrix; BA is not defined
CAB is defined and produces a 4×4 matrix; BA produces a 2×3 matrix
DNeither AB nor BA is defined because the outer dimensions don't match
AB requires A's columns to equal B's rows: A is 3×4 and B is 4×2, so the inner dimensions both equal 4 — AB is defined and produces a 3×2 matrix. For BA, B is 4×2 and A is 3×4: B's columns (2) must equal A's rows (3), which they don't — BA is undefined. This illustrates non-commutativity concretely: even the question of whether a product is defined depends on order.
Question 2 Multiple Choice
You apply transformation B to a vector, then apply transformation A to the result. Which matrix product encodes this composition?
AThe product BA, because you apply B first and A second, so B comes first in the expression
BThe product AB, because matrix multiplication is commutative and order doesn't matter
CThe product AB, because in function composition notation the rightmost matrix is applied first
DYou need two separate matrices; a single product cannot encode a two-step transformation
In matrix multiplication, the rightmost matrix is applied first — this is function composition notation. If you first apply B then A, the composition is written AB (read right-to-left: B acts first, A acts second). This is why order matters: AB and BA represent different sequences of transformations. Option A is the classic confusion — it reads left-to-right rather than following the composition convention.
Question 3 True / False
Matrix multiplication is commutative: for any two square matrices A and B of the same size, AB = BA.
TTrue
FFalse
Answer: False
Matrix multiplication is NOT commutative — even for square matrices of the same size, AB and BA are generally not equal. The geometric reason: applying transformation B then A typically gives a different result than applying A then B. For example, rotating then reflecting a shape is different from reflecting then rotating. This is one of the most important ways matrices differ from ordinary numbers.
Question 4 True / False
If AB is defined, then BA is expected to also be defined.
TTrue
FFalse
Answer: False
AB is defined when A's column count equals B's row count. If A is m×n and B is n×p, then AB is m×p. For BA to be defined, B's columns (p) must equal A's rows (m) — only guaranteed when m = p. For instance, a 2×3 matrix times a 3×5 matrix is defined (giving 2×5), but the reverse product (3×5)(2×3) requires 5 = 2, which is false — undefined.
Question 5 Short Answer
Explain why matrix multiplication is not commutative, using the connection between matrices and linear transformations.
Think about your answer, then reveal below.
Model answer: Matrix multiplication represents function composition of linear transformations. Composing transformations in different orders generally produces different results — rotating an object 90° then reflecting it gives a different final orientation than reflecting first then rotating. Since AB means 'apply B first, then A,' and BA means 'apply A first, then B,' the results are generally different. The non-commutativity of matrix multiplication is simply the non-commutativity of function composition made algebraic.
Students often expect multiplication to commute because ordinary number multiplication does. But matrices represent operations on space, not quantities, and the order in which operations are applied changes the outcome. This is why AB = BA almost never holds unless A and B have a special relationship.