RG's E-Learning Centre · Book a free demo class · WhatsApp +91-9379416374

Inverse of a 2×2 matrix: Class 12 practice

For A = [a b; c d] with |A| = ad − bc ≠ 0, A⁻¹ = (1/|A|)[d −b; −c a]. Read the method, follow the three worked examples, then try the six questions.

What you should know first

  • Determinants

The idea behind inverse of a 2×2 matrix

For A = [a b; c d] with |A| = ad − bc ≠ 0, A⁻¹ = (1/|A|)[d −b; −c a]. Check by multiplying: AA⁻¹ = I.

Step-by-step method

  1. Find |A|; if 0, no inverse.
  2. Swap a and d, change signs of b and c.
  3. Divide every entry by |A|.
  4. Check AA⁻¹ = I.

Worked example 1 (Basic)

Find the inverse of [-4 5; -3 6].

  1. |A| = -4×6 − 5×-3 = -9
  2. adj A = [6 -5; 3 -4]
  3. A⁻¹ = [-2/3 5/9; -1/3 4/9]
  4. Answer: [-2/3 5/9; -1/3 4/9]

Worked example 2 (Standard)

Find the inverse of [6 -3; 3 -5].

  1. |A| = 6×-5 − -3×3 = -21
  2. adj A = [-5 3; -3 6]
  3. A⁻¹ = [5/21 -1/7; 1/7 -2/7]
  4. Answer: [5/21 -1/7; 1/7 -2/7]

Worked example 3 (Challenge)

Find the inverse of [6 6; 1 2].

  1. |A| = 6×2 − 6×1 = 6
  2. adj A = [2 -6; -1 6]
  3. A⁻¹ = [1/3 -1; -1/6 1]
  4. Answer: [1/3 -1; -1/6 1]

Common mistakes and misconceptions

  • Forgetting to divide by |A|
  • Changing signs of a and d

Practice questions

  1. (Basic) Find the inverse of [6 -4; -5 -4].
  2. (Basic) Find the inverse of [6 1; 5 6].
  3. (Standard) Find the inverse of [1 -1; 5 2].
  4. (Standard) Find the inverse of [3 -1; 0 -1].
  5. (Challenge) Find the inverse of [-3 4; 1 6].
  6. (Challenge) Find the inverse of [-4 0; 1 1].

Answer key with full working

  • Q1: |A| = 6×-4 − -4×-5 = -44 → adj A = [-4 4; 5 6] → A⁻¹ = [1/11 -1/11; -5/44 -3/22] → Answer: [1/11 -1/11; -5/44 -3/22]
  • Q2: |A| = 6×6 − 1×5 = 31 → adj A = [6 -1; -5 6] → A⁻¹ = [6/31 -1/31; -5/31 6/31] → Answer: [6/31 -1/31; -5/31 6/31]
  • Q3: |A| = 1×2 − -1×5 = 7 → adj A = [2 1; -5 1] → A⁻¹ = [2/7 1/7; -5/7 1/7] → Answer: [2/7 1/7; -5/7 1/7]
  • Q4: |A| = 3×-1 − -1×0 = -3 → adj A = [-1 1; 0 3] → A⁻¹ = [1/3 -1/3; 0 -1] → Answer: [1/3 -1/3; 0 -1]
  • Q5: |A| = -3×6 − 4×1 = -22 → adj A = [6 -4; -1 -3] → A⁻¹ = [-3/11 2/11; 1/22 3/22] → Answer: [-3/11 2/11; 1/22 3/22]
  • Q6: |A| = -4×1 − 0×1 = -4 → adj A = [1 0; -1 -4] → A⁻¹ = [-1/4 0; 1/4 1] → Answer: [-1/4 0; 1/4 1]

Syllabus note

This skill is listed under "Determinants" 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 inverse of a 2×2 matrix in?

"Determinants", 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.

Book a free demo class