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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete answer

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

Definition

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.

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.

Rule to remember

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.

Memory hook

Matrices remember order; swapping factors changes the instruction.

Examples and method

Worked example

Take A = [[1, 2], [0, 1]] and B = [[3, 0], [1, 2]]. AB = [[1 x 3 + 2 x 1, 1 x 0 + 2 x 2], [0 x 3 + 1 x 1, 0 x 0 + 1 x 2]] = [[5, 4], [1, 2]]. BA = [[3 x 1 + 0 x 0, 3 x 2 + 0 x 1], [1 x 1 + 2 x 0, 1 x 2 + 2 x 1]] = [[3, 6], [1, 4]]. Hence AB != BA.

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

If AB exists, must BA also exist?

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

Answer writing and exam use

1-mark answer

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-mark answer

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. 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. 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-mark answer

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. 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. Take A = [[1, 2], [0, 1]] and B = [[3, 0], [1, 2]]. AB = [[1 x 3 + 2 x 1, 1 x 0 + 2 x 2], [0 x 3 + 1 x 1, 0 x 0 + 1 x 2]] = [[5, 4], [1, 2]]. BA = [[3 x 1 + 0 x 0, 3 x 2 + 0 x 1], [1 x 1 + 2 x 0, 1 x 2 + 2 x 1]] = [[3, 6], [1, 4]]. Hence AB != BA. Often asked through counterexamples, assertion-reason items, or simplification of expressions such as A(B+C) and (AB)C. Assuming AB = BA because multiplication of real numbers is commutative.
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