Q.If $A$ and $B$ are two square matrices of the same order, then $AB = BA$.

The key idea is that matrix multiplication is not commutative — the statement $AB = BA$ is false in general.

• For two square matrices of the same order, $AB$ and $BA$ are both defined and have the same size, but their entries are computed differently.
• The product $AB$ uses rows of $A$ with columns of $B$; $BA$ uses rows of $B$ with columns of $A$. These are not guaranteed to be equal. …

The heart of this question is a common misunderstanding: because numbers commute under multiplication ($3 \times 5 = 5 \times 3$), students often assume matrices do too. But matrices represent transformations (like rotations, reflections, scalings), and the order in which you apply two transformations usually matters.

Think of it this way:

• Rotating a shape then reflecting it gives a different result than reflecting then rotating.
• Matrix multiplication encodes that order-dependence. So $AB$ and $BA$ are generally different.

1. The claim is false for most square matrices. Take any two $2 \times 2$ matrices that are not scalar multiples of each other. For example:

Compute $AB$:

Compute $BA$:

Clearly $AB \neq BA$.

2. When does $AB = BA$ happen?

   Only in very restricted situations:

   • If $A$ or $B$ is the identity matrix $I$ (or a scalar multiple $kI$).
   • If both $A$ and $B$ are diagonal matrices.
   • If $A$ and $B$ are simultaneously diagonalizable (e.g., both polynomials in the same matrix).
   • If one is the zero matrix.

   But these are exceptions, not the rule. …

Method: Disproving a Universal Multiplication Claim by Counterexample

Use this when a statement claims a multiplication identity holds "for all square matrices of the same order." A single well-chosen counterexample is enough to disprove a universal claim.

Steps

Step 1: Note both products are defined and same-sized.

For square matrices of order $n$, both $AB$ and $BA$ exist and are $n\times n$, so size cannot settle it — you must compare entries.

Step 2: Pick the simplest possible test matrices.

Choose $2\times2$ matrices with mostly zeros and a single $1$, so the arithmetic is trivial and any difference is obvious. …

Common Mistakes

Mistake 1: Believing "same order" guarantees commutativity.

Why it's wrong: equal size only ensures both products exist and share a shape; the entries still differ. Correct approach: disprove with a $2\times2$ counterexample.

Mistake 2: Trying to "prove" it and forcing an error. …