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Matrix multiplication: Class 12 practice

AB is defined when the number of columns of A equals the number of rows of B. Read the method, follow the three worked examples, then try the six questions.

What you should know first

  • Integer arithmetic

The idea behind matrix multiplication

AB is defined when the number of columns of A equals the number of rows of B. Each entry (AB)ᵢⱼ is row i of A dotted with column j of B. In general AB ≠ BA.

Step-by-step method

  1. Check the orders are compatible.
  2. Multiply row by column and add.
  3. Write each result in its position.
  4. Do not assume AB = BA.

Worked example 1 (Basic)

Find AB for A = [3 2; 3 1] and B = [-1 -1; -3 3].

  1. Row 1 × columns: -9, 3
  2. Row 2 × columns: -6, 0
  3. Answer: [-9 3; -6 0]

Worked example 2 (Standard)

Find AB for A = [-3 -1; 3 4] and B = [2 1; 3 -1].

  1. Row 1 × columns: -9, -2
  2. Row 2 × columns: 18, -1
  3. Answer: [-9 -2; 18 -1]

Worked example 3 (Challenge)

For A = [4 -3; -3 -1] and B = [3 3; -2 1], find AB and BA. Are they equal?

  1. AB = [18 9; -7 -10]
  2. BA = [3 -12; -11 5]
  3. Not equal — matrix multiplication is not commutative
  4. Answer: AB = [18 9; -7 -10], BA = [3 -12; -11 5]

Common mistakes and misconceptions

  • Multiplying matching entries element by element
  • Assuming AB = BA

Practice questions

  1. (Basic) Find AB for A = [-3 -3; 3 -1] and B = [3 2; 2 -2].
  2. (Basic) Find AB for A = [0 4; -1 -2] and B = [2 -1; 0 0].
  3. (Standard) Find AB for A = [-3 1; -2 2] and B = [1 1; -2 1].
  4. (Standard) Find AB for A = [3 -1; -2 -3] and B = [3 -1; 2 1].
  5. (Challenge) For A = [-3 3; -2 0] and B = [-2 3; 2 4], find AB and BA. Are they equal?
  6. (Challenge) For A = [-3 0; 3 2] and B = [2 -2; -2 -2], find AB and BA. Are they equal?

Answer key with full working

  • Q1: Row 1 × columns: -15, 0 → Row 2 × columns: 7, 8 → Answer: [-15 0; 7 8]
  • Q2: Row 1 × columns: 0, 0 → Row 2 × columns: -2, 1 → Answer: [0 0; -2 1]
  • Q3: Row 1 × columns: -5, -2 → Row 2 × columns: -6, 0 → Answer: [-5 -2; -6 0]
  • Q4: Row 1 × columns: 7, -4 → Row 2 × columns: -12, -1 → Answer: [7 -4; -12 -1]
  • Q5: AB = [12 3; 4 -6] → BA = [0 -6; -14 6] → Not equal — matrix multiplication is not commutative → Answer: AB = [12 3; 4 -6], BA = [0 -6; -14 6]
  • Q6: AB = [-6 6; 2 -10] → BA = [-12 -4; 0 -4] → Not equal — matrix multiplication is not commutative → Answer: AB = [-6 6; 2 -10], BA = [-12 -4; 0 -4]

Syllabus note

This skill is listed under "Matrices" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

Related practice and notes

Frequently asked questions

Which chapter is matrix multiplication in?

"Matrices", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.

How were the answers checked?

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

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