Why Is Matrix Multiplication Non-Commutative?

Matrix multiplication is non-commutative because changing the order changes which rows are combined with which columns. Therefore, for matrices $A$ and $B$, generally $AB \neq BA$; sometimes one product may not even be defined.

This explanation assumes IB Mathematics: Applications and Interpretation HL, particularly AHL 1.14, Introduction to matrices, in a short-response Paper 1 style. A GDC is permitted, although a $2 \times 2$ product may be calculated by hand.

Why Order Matters

Each entry in a product is obtained using row-by-column multiplication. In $AB$, rows of $A$ multiply columns of $B$; in $BA$, rows of $B$ multiply columns of $A$. These are different calculations.

Consider

$$A=\begin{pmatrix}1&2\\0&1\end{pmatrix},\qquad B=\begin{pmatrix}1&0\\3&1\end{pmatrix}.$$

Hence, $AB \neq BA$.

Matrices can also represent transformations. In $AB\mathbf{x}$, matrix $B$ transforms $\mathbf{x}$ first, followed by $A$. In $BA\mathbf{x}$, the order is reversed. Just as rotating and then reflecting an object can differ from reflecting and then rotating it, the resulting matrices can differ.

For example, take the vector $\mathbf{x}$. Applying $B$ first gives $B\mathbf{x}$, and then applying $A$ gives $AB\mathbf{x}$. Reversing the order gives $A\mathbf{x}$ followed by . Since the final vectors can differ, the composed transformations differ. This is the conceptual reason that multiplication order is read from right to left in a matrix expression.

Dimensions provide another reason. If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ exists and is . However, exists only when , so reversing the order may be impossible.

Non-commutative does not mean that products are always unequal. A matrix commutes with itself, with a compatible identity matrix, and sometimes with other special matrices. However, commutativity must be demonstrated; it cannot be assumed. For an exam comparison, calculate both products explicitly before stating a conclusion.

Exam Technique

A common misconception is importing the commutative rule for numbers into matrices. In an IB response, show row-by-column working, check dimensions, and compare both products. Do not claim matrices never commute: special pairs, such as $AI=IA=A$, do commute.