Matrix Multiplication Explained: Rules, Steps and Examples | Class 12 CBSE

Matrix multiplication is one of the most important — and initially surprising — operations in Chapter 3. Unlike addition, where you simply add corresponding elements, multiplication requires a specific condition to hold and follows a very different process. It also breaks the familiar rule of commutativity that applies to numbers.

This guide explains matrix multiplication from the ground up: the condition for it to be defined, the step-by-step method, the properties you must know, and the key non-commutativity insight that separates matrix algebra from ordinary arithmetic.

What is Matrix Multiplication?

Matrix multiplication is the process of combining two matrices to produce a third, where each entry of the result is computed as the dot product of a row from the first matrix and a column from the second.

The Real-World Motivation

Imagine Meera wants to buy 2 pens and 5 story books, while Nadeem wants 8 pens and 10 story books. A shop quotes: pen = ₹5, story book = ₹50.

• Meera spends: 5×2 + 50×5 = ₹260

• Nadeem spends: 5×8 + 50×10 = ₹540

In matrix form:

Requirements:   | 2   5  |      Prices:  |  5 |
                | 8  10  |               | 50 |

Money needed:   | 5×2 + 50×5  |   =   | 260 |
                | 5×8 + 50×10 |         | 540 |

This is matrix multiplication: the requirements matrix times the price matrix gives the money needed.

Condition for Matrix Multiplication

The product AB is defined if and only if the number of columns in A equals the number of rows in B.

If A is m × n and B is n × p, then the product AB is a matrix of order m × p.

A       ×      B       =      AB
m × n       n × p          m × p
      ↑         ↑
   must match

If this inner dimension doesn’t match, the product is undefined.

How to Multiply Two Matrices: Step-by-Step

The element cᵢₖ of the product matrix C = AB is computed by:

cik = ai1·b1k + ai2·b2k + ... + ain·bnk  =  Σ aij · bjk

That is: take the i-th row of A and the k-th column of B, multiply corresponding elements, and sum the results.

Worked Example (NCERT Example 12)

Let A = [[6, 9], [2, 3]] and B = [[2, 6, 0], [7, 9, 8]]

A is 2 × 2, B is 2 × 3 → AB will be 2 × 3.

AB = | 6×2 + 9×7   6×6 + 9×9   6×0 + 9×8 |
     | 2×2 + 3×7   2×6 + 3×9   2×0 + 3×8 |

   = | 12+63   36+81   0+72 |   =  | 75   117   72 |
     |  4+21   12+27   0+24 |      | 25    39   24 |

Always confirm the order of the result before you compute: if A is m × n and B is n × p, your result must have exactly m rows and p columns.

Non-Commutativity of Matrix Multiplication

Here is the key difference from ordinary arithmetic: AB ≠ BA in general, even when both products are defined.

Example Demonstrating AB ≠ BA

Let A = [[1, -2, 3], [-4, 2, 5]] and B = [[2, 3], [4, 5], [2, 1]]

A is 2 × 3, B is 3 × 2. AB is 2 × 2 but BA is 3 × 3. Since AB and BA have different orders, they cannot be equal.

Even when both products have the same order, equality is not guaranteed. For example, with A = [[1, 0], [0, -1]] and B = [[0, 1], [1, 0]]:

AB = | 0   1 |    but    BA = |  0  -1 |
     | -1  0 |               |  1   0 |

AB ≠ BA. Matrix multiplication is not commutative — this is a fundamental and frequently tested fact.

When Is AB = BA?

AB = BA when A and B are diagonal matrices of the same order (the special case). In general, do not assume commutativity.

Zero Matrix as a Product of Non-Zero Matrices

In ordinary arithmetic, if ab = 0 then either a = 0 or b = 0. This is NOT true for matrices.

Example: Let A = [[0, -1], [0, 2]] and B = [[3, 5], [0, 0]]

AB = | 0×3 + (-1)×0   0×5 + (-1)×0 |   =   | 0  0 |   = O
     | 0×3 +  2×0     0×5 +  2×0   |        | 0  0 |

Neither A nor B is the zero matrix, yet AB = O. Remember this when evaluating matrix equations.

Properties of Matrix Multiplication

1. Associative Law

(AB)C = A(BC) whenever both sides are defined.

2. Distributive Law

A(B + C) = AB + AC and (A + B)C = AC + BC

3. Multiplicative Identity

For every square matrix A of order n: IₙA = AIₙ = A

The identity matrix acts as the “1” of matrix multiplication.

Using Matrix Multiplication to Solve Real Problems

Matrix multiplication is not an abstract exercise. In NCERT Example 19, a political group contacts voters in two cities using telephone, house calls, and letters. The cost per contact (in paise) forms a column matrix, and the number of contacts in each city forms a row matrix. Their product gives the total cost per city — a compact, elegant calculation.

Similarly, a school bookshop with three book types at three prices uses a row matrix of quantities and a column matrix of prices. One matrix product gives the total revenue — replacing the need for three separate multiplications and an addition. This is the power of matrix multiplication: it bundles multiple simultaneous calculations into one clean operation.

CBSE Board Exam Tips

• Check dimensions first: Write A(m×n) × B(n×p) → AB(m×p) before computing.

• Non-commutativity: Never swap the order of A and B. AB and BA are generally different.

• Row-column method: For each element cᵢₖ, dot product row i of A with column k of B.

• Zero product: The fact that AB = O does not mean A = O or B = O.

• Identity: Multiplying any matrix by the identity matrix of the right order returns the original.

Suggested Reading on ChampionsPrep

• Matrix Addition and Scalar Multiplication: Operations and Properties

• Transpose of a Matrix: Properties and the Reversal Law for Class 12

• Invertible Matrices: Definition, Uniqueness and (AB)⁻¹ Theorem

Test Your Knowledge

Test Your Matrix Multiplication Skills

Multiplication is the trickiest operation in this chapter — and the most frequently tested. Speed and accuracy come with practice.

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