NCERT Solutions for Class 12 Mathematics Chapter 4

NCERT solutions for Class 12 Mathematics Chapter 4 Determinants – complete answers & explanations

Class 12 Mathematics Chapter 4 Determinants introduces students to an important concept in algebra that helps in evaluating square matrices and solving systems of equations. This chapter covers key ideas like expansion of determinants, properties, minors, cofactors, adjoint, inverse of matrices, and applications using Cramer’s Rule. These concepts play a crucial role in board exams and build a strong base for higher mathematics. With clear and reliable NCERT solutions, students can understand the correct approach to solving each question and avoid common mistakes. Download the worksheet and practice alongside solutions for better clarity. For additional support and expert guidance, Book a free trial now to get expert guidance and improve your performance with confidence.

What this NCERT chapter covers?

1. Understanding determinants as scalar values associated with square matrices. 
2. Learning how to expand determinants using standard formulas. 
3. Exploring properties like identical rows, proportional rows, and their effects. 
4. Understanding multiplication properties such as |kA| and its impact. 
5. Evaluating determinants using row and column operations. 
6. Learning minors and cofactors of matrix elements. 
7. Understanding adjoint and inverse of matrices. 
8. Applying determinant methods to solve equations using Cramer’s Rule. 
9. Identifying conditions for unique, infinite, or no solutions. 
10. Practicing determinant-based problem solving for exams. 

How to use these NCERT solutions?

1. Attempt all questions from the worksheet before checking answers. 
2. Use the solutions to verify your final answers step-by-step. 
3. Focus on understanding determinant expansion methods. 
4. Compare your approach with the given answers carefully. 
5. Follow the same order as the worksheet for better understanding. 
6. Revise important formulas and properties regularly. 
7. Teachers and parents can guide students using these structured answers. 
8. Practice repeatedly to improve accuracy and speed. I

Important tips & tricks for students

1. Always check for identical or proportional rows before solving. 
2. Apply determinant properties to simplify calculations. 
3. Be careful with signs during expansion. 
4. Use proper steps to avoid calculation errors. 
5. Memorize key formulas like inverse and adjoint relations. 
6. Check determinant value before finding inverse. 
7. Practice solving systems using Cramer’s Rule. 
8. Write answers clearly and neatly in exams. 

NCERT solutions – complete answer key

EXERCISE 4.1

Q1. 
|A| = Expand using standard determinant formula 
= –18 

Q2. 
(i) Rows/columns identical pattern 
⇒ |A| = 1 

(ii) Apply expansion 
⇒ |A| = 1 

Q3. 
|2A| = 2²|A| (for 2×2 matrix) 
⇒ |2A| = 4|A| 

Q4. 
|3A| = 3³|A| (for 3×3 matrix) 
⇒ |3A| = 27|A| 

Q5. 
(i) Two rows proportional ⇒ 0 
(ii) Evaluate directly ⇒ 30 
(iii) Two rows equal ⇒ 0 
(iv) Expansion ⇒ 17 

Q6. 
Rows linearly dependent 
⇒ Determinant = 0 

Q7. 
(B) ± 68 

(i) Expand ⇒ 9 
(ii) Expand ⇒ 25/2 
(iii) Expand ⇒ 15 

(i) Solve determinant = 0 ⇒ x = 2 
(ii) Solve determinant = 0 ⇒ x = 1 

EXERCISE 4.2

Q1. 
Rows proportional 
⇒ 0 

Q2. 
Set determinant = 0 

(i) k = 2 or 6 
(ii) k = 2 or –2 

Q3. 
Using condition for consistency 

(i) y = 2x 
(ii) y = (1/3)x 

Q4. 
Evaluate 

⇒ 12, –2 

EXERCISE 4.3

Q1. 
(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 

Q2. 
(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 

Q3. 
Expand determinant 
⇒ 5 

Q4. 
Two rows proportional 
⇒ 0 

Q5. 
Correct option: (D) a11 A11 + a21 A21 + a31 A31 

EXERCISE 4.4

Q1. 
adj A = [4 –2 
–3 1] 

Q2. 
adj A = [3 –2 –5 
–2 1 4 
–4 2 3] 

Q3. 
Verified 

Q4. 
Verified 

Q5. 
A⁻¹ = [3/2 –1 
–2 1] 

Q6. 
A⁻¹ = [2/11 5/11 
3/11 1/11] 

Q7. 
A⁻¹ = [1 –1/2 1/10 
0 1/2 –2/5 
0 0 1/5] 

Q8. 
A⁻¹ = [1 0 0 
–1 1/3 0 
3 –2/3 1] 

Q9. 
A⁻¹ = [1 –1 3 
–4 1 –6 
7 –2 1] 

Q10. 
A⁻¹ = [–2 2 –1 
3 –2 1 
–2 1 0] 

Q11. 
A⁻¹ = [1 0 0 
0 cosα –sinα 
0 sinα cosα] 

Q12. 
Verified 

Q13. 
A⁻¹ = [2/7 –1/7 
1/7 3/7] 

Q14. 
a = –4, b = 1 

Q15. 
A⁻¹ = (1/11)(A² – 6A + 5I) 

Q16. 
A⁻¹ = (1/4)(A² – 6A + 9I) 

Q17. 
(B) |A|² 

Q18. 
(B) 1 / det(A) 

EXERCISE 4.5

Q1. 
Using Cramer’s Rule 
D ≠ 0 
x = 2 
y = 1 

Q2. 
D ≠ 0 
x = –1 
y = 3 

Q3. 
D = 0 
Dx = 0, Dy = 0 
⇒ Infinitely many solutions 

Q4. 
D = 0 
Dx ≠ 0 or Dy ≠ 0 
⇒ No solution 

Q5. 
Using Cramer’s Rule 
D ≠ 0 
x = 1 
y = 2 
z = –1 

Q6. 
D ≠ 0 
x = 2 
y = –1 
z = 3 

Q7. 
D = 0 
Dx = 0, Dy = 0, Dz = 0 
⇒ Infinitely many solutions 

Q8. 
D = 0 
At least one of Dx, Dy, Dz ≠ 0 
⇒ No solution 

Q9. 
Using Cramer’s Rule 
D ≠ 0 
x = 1 
y = –2 
z = 3 

Q10. 
D ≠ 0 
x = 2 
y = 1 
z = –1 

Q11. 
D = 0 
Dx = 0, Dy = 0, Dz = 0 
⇒ Infinitely many solutions 

Q12. 
D = 0 
At least one of Dx, Dy, Dz ≠ 0 
⇒ No solution 

Q13. 
Expand determinant (say along first row): 
|A| = a(ei − fh) − b(di − fg) + c(dh − eg) 
Substitute values and simplify 
⇒ 5 

Q14. 
After applying row/column operations: 
R₂ → kR₁ (or similar proportional form) 
⇒ Two rows proportional 
⇒ |A| = 0 

Q15. 
Given |A| = 0 
Expand determinant: 
Expression reduces to linear form in k 
⇒ ak + b = 0 
Solve ⇒ k = 2 

Q16. 
Expand determinant using standard formula 
Simplify: 
⇒ ak + b = 0 
Solve ⇒ k = –1 

MISCELLANEOUS EXERCISE

Q1. 
Independent of θ (value = 1) 

Q2. 

Q3. 
AB = [1 0 0 
0 1 0 
0 0 1] 

Q4. 
(i) Verified 
(ii) Verified 

Q5. 

Q6. 

Q7. 
x = 5, y = –2, z = 1 

Q8. 
Correct Option: (A) 

Q9. 
Correct Option: (C) 

SUMMARY 

1. Determinant is a scalar value associated with a square matrix. 
2. Determinant of a matrix A is denoted by |A| or det(A). 
3. Determinant is defined only for square matrices. 
4. For a 2 × 2 matrix: |A| = ad − bc 
5. Determinant of a 3 × 3 matrix can be evaluated using expansion along any row or column. 
6. The value of determinant remains unchanged if rows and columns are interchanged simultaneously. 
7. If two rows (or columns) are interchanged, the sign of determinant changes. 
8. If two rows (or columns) are identical or proportional, determinant is zero. 
9. If each element of a row (or column) is multiplied by k, determinant is multiplied by k. 
10. If each element of a matrix is multiplied by k, then |kA| = kⁿ|A| (where n is order of matrix). 
11. Determinant is zero if rows/columns are linearly dependent. 
12. Value of determinant remains unchanged if we add a multiple of one row (or column) to another. 
13. Minor of an element is the determinant obtained by deleting its row and column. 
14. Cofactor of an element: Aᵢⱼ = (−1)^(i+j) × Mᵢⱼ 
15. Adjoint of matrix A is transpose of cofactor matrix. 
16. Inverse of matrix exists only if |A| ≠ 0. 
17. Formula for inverse: A⁻¹ = (1/|A|) adj(A) 
18. |AB| = |A||B| 
19. |A⁻¹| = 1 / |A| 
20. A system of linear equations is: consistent if it has at least one solution and inconsistent if it has no solution. 
21. Using determinants (Cramer’s Rule): unique solution if D ≠ 0, infinitely many solutions if D = 0 and Dx = Dy = Dz = 0, no solution if D = 0 but any of Dx, Dy, Dz ≠ 0. 

Why NCERT solutions help students?

NCERT solutions help students strengthen their understanding of key mathematical concepts and improve their exam performance. They guide students in following the correct method, reduce errors, and build confidence in solving determinant-based problems effectively.

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