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Caveats: Non-Commutativity and Zero Products

Non-commutativity: $AB \neq BA$ in general, even for square matrices. A simple example: rotate then scale is different from scale then rotate (if scales are non-uniform). This is the most important difference from scalar arithmetic.

Zero products: For scalars, $ab = 0$ implies $a = 0$ or $b = 0$ . For matrices this fails: $AB$ can be the zero matrix with $A \neq 0$ and $B \neq 0$ . This happens when the columns of $B$ all lie in the null space of $A$ .

No cancellation: $AB = AC$ does NOT imply $B = C$ (unless $A$ is invertible). Similarly $BA = CA$ does not imply $B = C$ .

Formal View

Example 4.6 — Zero Product of Nonzero Matrices

A = \begin{pmatrix}0 & 1\\0 & 0\end{pmatrix}

B = \begin{pmatrix}1 & 0\\0 & 0\end{pmatrix}

. Then

AB = \begin{pmatrix}0 & 0\\0 & 0\end{pmatrix}

even though

A \neq 0

and

B \neq 0

Why This Matters

These failures of familiar algebra cause subtle bugs in matrix computations and derivations.

In programming: matrix product order in shader code critically affects 3D rendering
In quantum mechanics: operator order (commutators) determines observable properties
Numerical: accumulation of rounding errors differs between $(AB)C$ and $A(BC)$ despite mathematical equality

Learning Resources

Non-commutativity of matrices

Khan Academy

Why $AB \neq BA$ and examples.

11 min

Quiz

Question 1

If $AB = 0$ , then either $A = 0$ or $B = 0$ .

Common Mistakes

Applying scalar arithmetic rules ( $ab = ba$ , $ab = 0 \Rightarrow a = 0$ ) to matrices.
Canceling matrices from both sides of an equation without checking invertibility.