Adjoint and inverse of a 3×3 matrix: Class 12 practice

What you should know first

• 2×2 determinants

The idea behind adjoint and inverse of a 3×3 matrix

A⁻¹ = adj(A)/|A| when |A| ≠ 0. adj(A) is the transpose of the cofactor matrix. Check: A × A⁻¹ = I.

Step-by-step method

1. Find |A|; if 0, there is no inverse.
2. Find all nine cofactors.
3. Transpose to get adj A; divide by |A|.

Worked example 1 (Basic)

Find |A| for A = [1, 1, 0; 3, -1, 0; 1, 1, 1].

1. Expand along row 1
2. |A| = -4
3. Answer: -4

Worked example 2 (Standard)

Find adj A for A = [-2, 4, 0; 3, -1, -2; 1, 3, -2].

1. Cofactors [8, 4, 10; 8, 4, 10; -8, -4, -10]
2. Transpose
3. Answer: [8, 8, -8; 4, 4, -4; 10, 10, -10]

Worked example 3 (Challenge)

Find A⁻¹ for A = [1, 4, 1; 0, 0, 3; 4, -2, 3].

1. |A| = 54
2. adj A = [6, -14, 12; 12, -1, -3; 0, 18, 0]
3. Answer: (1/54) × [6, -14, 12; 12, -1, -3; 0, 18, 0]

Common mistakes and misconceptions

• Forgetting to transpose
• Sign pattern of cofactors

Practice questions

1. (Basic) Find |A| for A = [2, 4, -2; 3, 2, -1; -2, -1, 3].
2. (Basic) Find |A| for A = [-1, 2, -3; 4, -2, -3; -2, -2, 1].
3. (Standard) Find adj A for A = [2, 2, 3; 2, 3, 4; 1, -3, 4].
4. (Standard) Find adj A for A = [0, -1, 4; 2, 2, 1; 1, -2, -3].
5. (Challenge) Find A⁻¹ for A = [2, -2, -2; 1, 2, -3; -1, 0, -1].
6. (Challenge) Find A⁻¹ for A = [4, -1, 2; 4, 3, 1; 0, -2, 1].

Answer key with full working

• Q1: Expand along row 1 → |A| = -20 → Answer: -20
• Q2: Expand along row 1 → |A| = 48 → Answer: 48
• Q3: Cofactors [24, -4, -9; -17, 5, 8; -1, -2, 2] → Transpose → Answer: [24, -17, -1; -4, 5, -2; -9, 8, 2]
• Q4: Cofactors [-4, 7, -6; -11, -4, -1; -9, 8, 2] → Transpose → Answer: [-4, -11, -9; 7, -4, 8; -6, -1, 2]
• Q5: |A| = -16 → adj A = [-2, -2, 10; 4, -4, 4; 2, 2, 6] → Answer: (1/-16) × [-2, -2, 10; 4, -4, 4; 2, 2, 6]
• Q6: |A| = 8 → adj A = [5, -3, -7; -4, 4, 4; -8, 8, 16] → Answer: (1/8) × [5, -3, -7; -4, 4, 4; -8, 8, 16]

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

• Determinants chapter notes (Class 12)
• Evaluating 2×2 and 3×3 determinants
• Inverse of a 2×2 matrix
• Solving equations with matrices / Cramer's rule
• Derivatives of polynomials and slope of a tangent
• Definite integrals of polynomials
• Matrix multiplication
• Conditional probability and Bayes' theorem
• All Class 12 practice

Frequently asked questions

Which chapter is adjoint and inverse of a 3×3 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.