medium difficulty8 min

Properties of Matrix Multiplication

Matrix multiplication is associative and distributive wherever the products are defined, but it is not commutative in general; usually AB is not equal to BA.

In one line: Matrices remember order; swapping factors changes the instruction.

Practice This Concept

Understand it

For compatible matrices, A(BC) = (AB)C and A(B+C) = AB + AC. However, changing the order of multiplication can change the answer or may make the product undefined. Therefore, algebraic rearrangement must preserve order.

Example

If A = [[1, 2], [0, 1]] and B = [[3, 0], [1, 2]], then AB = [[5, 4], [1, 2]] while BA = [[3, 6], [1, 4]], so AB ≠ BA.

Formula and rule

Associative law: A(BC) = (AB)C when products are defined.

Distributive laws: A(B+C) = AB + AC and (A+B)C = AC + BC when orders allow.

Non-commutative rule: AB need not equal BA.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Take A = [[1, 2], [0, 1]] and B = [[3, 0], [1, 2]].

1AB = [[1 × 3 + 2 × 1, 1 × 0 + 2 × 2], [0 × 3 + 1 × 1, 0 × 0 + 1 × 2]] = [[5, 4], [1, 2]].
2BA = [[3 × 1 + 0 × 0, 3 × 2 + 0 × 1], [1 × 1 + 2 × 0, 1 × 2 + 2 × 1]] = [[3, 6], [1, 4]].
3Hence AB ≠ BA.

Method you can reuse

1Check whether each product exists.
2Keep matrix order unchanged.
3Apply associativity only to regroup brackets.
4Apply distributivity only when the addition inside the bracket is defined.
5Use a small counterexample when asked to disprove commutativity.

Visual tip

No diagram is needed; the concept depends on order and algebraic properties.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If AB exists, must BA also exist?

Think first, then show answer

No. For example, if A is 2 × 3 and B is 3 × 4, then AB exists and is 2 × 4, but BA is not defined because 4 and 2 do not match.

Common wrong answer

Assuming AB = BA because multiplication of real numbers is commutative.

Don't mix this up

Students cancel or rearrange matrices like ordinary numbers, for example changing AB to BA without justification.

Exam tip

Never change the order of factors unless a condition such as AB = BA is specifically given or proved.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Often asked through counterexamples, assertion-reason items, or simplification of expressions such as A(B+C) and (AB)C.

1
Enough for 1 mark
Matrix multiplication is associative and distributive wherever the products are defined, but it is not commutative in general; usually AB is not equal to BA.
2
Add for 2 marks
Example: If A = [[1, 2], [0, 1]] and B = [[3, 0], [1, 2]], then AB = [[5, 4], [1, 2]] while BA = [[3, 6], [1, 4]], so AB ≠ BA.
3
Add for 3 marks
For compatible matrices, A(BC) = (AB)C and A(B+C) = AB + AC.

MCQ Quiz

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