K2KARAKORAM
English
Open the app

CBSE · Class 12 · Mathematics

Multiplication of Matrices (Ch 3: 3.4.5–3.4.6)

2 concepts · 16 practice questions with answers · Free to read

Product of two matrices

From daily life

A shop has a price list for pens and books and a table of how many of each three customers bought. Multiplying the two tables gives each customer's bill.

Simple definition

AB is defined only if the number of columns of A equals the number of rows of B. If A is m × n and B is n × p then AB is m × p. The entry in row i, column j of AB is the sum of products of row i of A with column j of B.

Formula: (AB)ᵢⱼ = aᵢ₁ b₁ⱼ + aᵢ₂ b₂ⱼ + ... + aᵢₙ bₙⱼ

Rule 1: Multiply — Take a row of A and a column of B, multiply pairs, add.

  • Easy: [1 2 ; 3 4] [5 ; 6] = [1×5 + 2×6 ; 3×5 + 4×6] = [17 ; 39].
  • Medium: [1 2 ; 3 4] [5 6 ; 7 8] = [19 22 ; 43 50].
  • Exam-type: [1 0 2 ; 3 1 1] [2 1 ; 0 3 ; 1 0] = [4 1 ; 7 6].

Rule 2: Check the order first — Inner numbers must match; outer numbers give the answer.

  • Easy: (2 × 3)(3 × 4) gives a 2 × 4 matrix.
  • Medium: (3 × 1)(1 × 3) gives a 3 × 3 matrix.
  • Exam-type: (2 × 3)(2 × 3) is not defined because 3 is not 2.
Remember

Row of the first times column of the second, always in that order.

Common mistake

Multiplying entry by entry like addition; that is not matrix multiplication.

Memory clue

Inner numbers match, outer numbers survive: (m × n)(n × p) = m × p.

Practice questions

Easy
  1. 1A is 3 × 2 and B is 2 × 5. What is the order of AB?
  2. 2A is 2 × 3 and B is 2 × 3. Is AB defined?
  3. 3Find [2 3] [4 ; 5].
Medium
  1. 4Find [1 2 ; 0 1] [3 0 ; 1 2].
  2. 5Find [2 0 ; 1 3] [1 1 ; 2 0].
  3. 6For A = [1 2 ; 3 4] and B = [0 1 ; 1 0], find AB and BA.
Challenging
  1. 7Find [1 2 3] [2 ; −1 ; 1].
  2. 8Find A² for A = [1 −1 ; 2 0].
Show answers (try them yourself first)
  1. 13 × 5
  2. 2No, columns of A (3) differ from rows of B (2)
  3. 3[2×4 + 3×5] = [23]
  4. 4[5 4 ; 1 2]
  5. 5[2 2 ; 7 1]
  6. 6AB = [2 1 ; 4 3] and BA = [3 4 ; 1 2], so AB is not BA
  7. 7[2 − 2 + 3] = [3]
  8. 8A² = [−1 −1 ; 2 −2]

Properties of matrix multiplication

From daily life

Putting on socks and then shoes is not the same as shoes and then socks. Matrix multiplication also depends on the order.

Simple definition

Matrix multiplication is not commutative (AB may differ from BA), but it is associative: (AB)C = A(BC), and distributive: A(B + C) = AB + AC. For the identity, AI = IA = A. AB = O does not force A = O or B = O.

Formula: AI = IA = A, (AB)C = A(BC), AB ≠ BA in general

Rule 1: Show a property fails — One example is enough.

  • Easy: A = [1 0 ; 0 0], B = [0 1 ; 0 0]: AB = [0 1 ; 0 0] but BA = [0 0 ; 0 0].
  • Medium: A = [1 0 ; 0 0], B = [0 0 ; 0 1] give AB = O although A and B are not O.
  • Exam-type: (A + B)² = A² + AB + BA + B², which equals A² + 2AB + B² only when AB = BA.

Rule 2: Compute powers — Multiply step by step and look for a pattern.

  • Easy: A = [1 1 ; 0 1]: A² = [1 2 ; 0 1].
  • Medium: Then A³ = [1 3 ; 0 1], so Aⁿ = [1 n ; 0 1].
  • Exam-type: A = [0 1 ; 0 0] has A² = O.
Remember

Multiplying by I changes nothing; multiplying by O gives O.

Common mistake

Cancelling a common matrix factor, like turning AB = AC into B = C.

Memory clue

Order matters, grouping does not.

Practice questions

Easy
  1. 1Find A I for A = [3 5 ; 2 1].
  2. 2If AB = O for 2 × 2 matrices, must A = O or B = O?
  3. 3Find A² for A = [1 1 ; 0 1].
Medium
  1. 4Find A³ for A = [1 1 ; 0 1].
  2. 5Find A² for A = [0 1 ; 0 0].
  3. 6Find A² for A = [2 0 ; 0 3].
Challenging
  1. 7For A = [1 2 ; 0 1] and B = [1 0 ; 3 1], find AB and BA. Are they equal?
  2. 8For A = [1 2 ; 0 1], find A² − 2A + I.
Show answers (try them yourself first)
  1. 1A itself, [3 5 ; 2 1]
  2. 2No; for example [1 0 ; 0 0] [0 0 ; 0 1] = O
  3. 3[1 2 ; 0 1]
  4. 4[1 3 ; 0 1]
  5. 5O = [0 0 ; 0 0]
  6. 6[4 0 ; 0 9]
  7. 7AB = [7 2 ; 3 1] and BA = [1 2 ; 3 7]; they are not equal
  8. 8A² = [1 4 ; 0 1], 2A = [2 4 ; 0 2], so A² − 2A + I = [0 0 ; 0 0]