Q.Matrix multiplication is commutative.

Concept: Matrix Multiplication Compatibility — matrix multiplication is not commutative in general. For two matrices $A$ and $B$, $AB$ and $BA$ are defined only when the dimensions match, and even then they are rarely equal.

Reasoning:

1. For $AB$ to be defined, $A$ must be $m \times n$ and $B$ must be $n \times p$. For $BA$ to be defined, $B$ must be $p \times q$ and $A$ must be $q \times m$. These conditions are almost never simultaneously satisfied unless $m = n = p = q$ (square matrices of the same order). …

Why This Statement is Wrong

The idea that multiplication is commutative — that $AB = BA$ — is so deeply ingrained from arithmetic that it feels natural to carry it over to matrices. But matrices are not numbers. They represent transformations (like rotations, stretches, or projections), and the order in which you apply transformations matters enormously.

Think of it this way: putting on your socks then your shoes is not the same as putting on your shoes then your socks. Matrix multiplication is like composing functions — and function composition is never commutative in general.

Step-by-Step Reasoning

1. The definition of commutativity

For an operation $\ast$ to be commutative, we need $A \ast B = B \ast A$ for all possible $A$ and $B$ in the set. For matrix multiplication, this would mean $AB = BA$ for every pair of matrices where both products are defined.

2. The first obstacle: the products may not both exist

If $A$ is $m \times n$ and $B$ is $p \times q$, then $AB$ is defined only when $n = p$, and $BA$ is defined only when $q = m$. For both products to exist, we need $n = p$ and $m = q$ — meaning both matrices must be square and of the same size. So for non-square matrices, commutativity isn't even a meaningful question.

3. Even for square matrices of the same size, commutativity fails

Take a simple $2 \times 2$ example:

Compute $AB$:

Now compute $BA$:

Clearly $AB \neq BA$. One counterexample is enough to disprove the statement.

4. When does commutativity happen? …

Method: Testing a Claimed Matrix-Algebra Property (Prove or Disprove)

Use this whenever a statement asserts a general algebraic property of matrices ("$X$ is always true"). The single most common error is importing a rule from ordinary-number algebra without checking it survives in matrix algebra.

Steps

Step 1: State the property precisely and decide what would settle it.

A property claimed "for all matrices" is disproved by a single counterexample, but can only be proved by a general argument. So first ask: does this feel like a real theorem, or a number-algebra habit being carried over? Commutativity ($AB = BA$) is the classic habit.

Step 2: Check whether both sides are even defined.

For $AB$ and $BA$ to both exist with $A$ of order $m\times n$ and $B$ of order $p\times q$, you need $n=p$ (for $AB$) and $q=m$ (for $BA$). Unless both matrices are square of the same order, the two products differ in size and cannot be equal — that alone can settle it.

Step 3: If both are square, hunt for a small counterexample. …

Common Mistakes

Mistake 1: Assuming matrix multiplication behaves like number multiplication.

Why it's wrong: for numbers $ab=ba$ always, but a matrix product mixes rows of one with columns of the other, so $AB$ and $BA$ generally differ. Correct approach: treat commutativity as something to be checked, never assumed — produce a $2\times 2$ counterexample such as $A=\begin{bmatrix}1&1\\0&1\end{bmatrix}$, $B=\begin{bmatrix}1&0\\1&1\end{bmatrix}$, giving $AB\neq BA$.

Mistake 2: Thinking that if both $AB$ and $BA$ are defined they must be equal. …