Non-Commutativity and Zero-Divisors in Matrix Multiplication

Matrix Multiplication Is Not Commutative

Unlike addition of matrices and unlike ordinary multiplication of real numbers, matrix multiplication does not, in general, satisfy $AB=BA$ -- even when both products are defined and have the same order. This is proved, not merely asserted, by a single concrete counterexample.

Counterexample. Let $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ and $B=\begin{bmatrix}2&0\\1&3\end{bmatrix}$ (Example 5). Computing both products by the row-by-column rule of Section 5:

Comparing entry by entry, $AB\neq BA$ -- for instance the $(1,1)$ entries alone already differ ($4\neq2$). A single such example is enough to disprove commutativity as a general law, since a genuine algebraic law must hold for every pair of matrices, not just some. (Exercise: Matrix Multiplication, Q1--Q2 repeats this check with a fresh pair of matrices.)

Zero-Divisors: A Product Can Vanish Without Either Factor Vanishing

In ordinary arithmetic of real numbers, $ab=0$ forces $a=0$ or $b=0$ -- there is no way around it. Matrices behave completely differently: it is possible for neither factor to be the zero matrix, yet their product still to be the zero matrix.

Worked example. Let

Neither $A$ nor $B$ is the zero matrix ($A$ has a $1$ in position $(1,1)$; $B$ has a $1$ in position $(2,2)$). Yet

$AB=\begin{bmatrix}1\cdot0+0\cdot0 & 1\cdot0+0\cdot1\\0\cdot0+0\cdot0 & 0\cdot0+0\cdot1\end{bmatrix}=\begin{bmatrix}0&0\\0&0\end{bmatrix}=O.$ …