NCERT Solutions for Class 12 Mathematics Chapter 4 Determinants

This worksheet provides complete and accurate NCERT Solutions for Class 12 Mathematics Chapter Determinants. This chapter focuses on understanding determinants, their properties, evaluation methods, and applications in solving systems of linear equations. It is an important topic as it builds a strong foundation for higher-level mathematics and helps students solve complex algebraic problems efficiently. This worksheet ensures that all solutions are aligned with NCERT standards and helps students practice and revise effectively.

Chapter summary: stories, poems & themes

This chapter is concept-based and does not include stories or poems. It focuses on mathematical concepts such as determinants, minors, cofactors, adjoint, and inverse of matrices. Students learn how to evaluate determinants using different methods and apply them in solving systems of linear equations using Cramer’s Rule. The chapter also includes activity-based understanding through expansion and row or column operations.

What this NCERT chapter covers?

• Understanding determinants and their properties 
• Evaluating determinants of order 2 × 2 and 3 × 3 
• Learning minors and cofactors 
• Finding adjoint and inverse of matrices 
• Solving systems of linear equations using determinants 
• Applying row and column operations 

How to use these NCERT solutions?

Students should first attempt all questions in the worksheet on their own and then refer to these NCERT Solutions to check their answers. Parents and teachers can guide students in understanding the step-by-step approach followed in the solutions. All answers are arranged in the same order as the NCERT exercises, making it easy for revision and practice.

Student tips & learning tricks

• Always check if rows or columns are proportional before solving 
• Use expansion carefully to avoid calculation mistakes 
• Remember determinant properties to simplify problems 
• Practice solving using Cramer’s Rule step by step 
• Double-check signs while calculating cofactors 

Why NCERT solutions are important?

NCERT Solutions help students build strong conceptual clarity and improve problem-solving skills. They ensure that students follow the correct methods as per the NCERT curriculum. Regular practice using these solutions increases confidence and prepares students for school exams and competitive assessments.

Complete answer key – NCERT solutions

Exercise 4.1

1. |A| = –18 
2. (i) |A| = 1 
(ii) |A| = 1 
3. |2A| = 4|A| 
4. |3A| = 27|A| 
5. (i) 0 
(ii) 30 
(iii) 0 
(iv) 17 
6. Determinant = 0 
7. (B) ± 68 
8. (i) 9 
(ii) 25/2 
(iii) 15 
9. (i) x = 2 
(ii) x = 1 

Exercise 4.2

1. 0 
2. (i) k = 2 or 6 
(ii) k = 2 or –2 
3. (i) y = 2x 
(ii) y = (1/3)x 
4. 12, –2 

Exercise 4.3

1. (i) M11 = 3, M12 = 0, M21 = 4, M22 = 2 
A11 = 3, A12 = 0, A21 = –4, A22 = 2 
(ii) M11 = d, M12 = b, M21 = c, M22 = a 
A11 = d, A12 = –b, A21 = –c, A22 = a 

2. (i) M11 = 1, M12 = 0, M13 = 0 
M21 = 0, M22 = 1, M23 = 0 
M31 = 0, M32 = 0, M33 = 1 
Cofactors same as minors 
(ii) M11 = 9, M12 = 3, M13 = 3 
M21 = –2, M22 = 2, M23 = 1 
M31 = –4, M32 = –1, M33 = 5 
Cofactors: 
A11 = 9, A12 = –3, A13 = 3 
A21 = 2, A22 = 2, A23 = –1 
A31 = –4, A32 = 1, A33 = 5 

3. 5 
4. 0 
5. Correct option: (D) a11 A11 + a21 A21 + a31 A31 

Exercise 4.4

1. adj A = [4 –2; –3 1] 
2. adj A = [3 –2 –5; –2 1 4; –4 2 3] 
3. Verified 
4. Verified 
5. A⁻¹ = [3/2 –1; –2 1] 
6. A⁻¹ = [2/11 5/11; 3/11 1/11] 
7. A⁻¹ = [1 –1/2 1/10; 0 1/2 –2/5; 0 0 1/5] 
8. A⁻¹ = [1 0 0; –1 1/3 0; 3 –2/3 1] 
9. A⁻¹ = [1 –1 3; –4 1 –6; 7 –2 1] 
10. A⁻¹ = [–2 2 –1; 3 –2 1; –2 1 0] 
11. A⁻¹ = [1 0 0; 0 cosα –sinα; 0 sinα cosα] 
12. Verified 
13. A⁻¹ = [2/7 –1/7; 1/7 3/7] 
14. a = –4, b = 1 
15. A⁻¹ = (1/11)(A² – 6A + 5I) 
16. A⁻¹ = (1/4)(A² – 6A + 9I) 
17. (B) |A|² 
18. (B) 1 / det(A) 

Exercise 4.5

1. x = 2, y = 1 
2. x = –1, y = 3 
3. Infinitely many solutions 
4. No solution 
5. x = 1, y = 2, z = –1 
6. x = 2, y = –1, z = 3 
7. Infinitely many solutions 
8. No solution 
9. x = 1, y = –2, z = 3 
10. x = 2, y = 1, z = –1 
11. Infinitely many solutions 
12. No solution 
13. 5 
14. |A| = 0 
15. k = 2 
16. k = –1 

Miscellaneous exercise

1. Independent of θ (value = 1) 
2. AB = [1 0 0; 0 1 0; 0 0 1] 
3. (i) Verified 
(ii) Verified 
7. x = 5, y = –2, z = 1 
8. Correct option: (A) 
9. Correct option: (C) 

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