Determinants Class 12 Notes

• |Aᵀ| = |A| — transposing does not change the determinant.
• Interchanging two rows or columns changes the sign of the determinant.
• If two rows or columns are identical or proportional, the determinant is 0.
• A row can be scaled by multiplying the whole determinant by that factor.
• Adding a multiple of one row to another leaves the determinant unchanged.

• For a₁x + b₁y + c₁z = d₁, a₂x + b₂y + c₂z = d₂, a₃x + b₃y + c₃z = d₃, compute D = |coefficient matrix|.
• x = Dₓ/D, y = D_y/D, z = D_z/D where Dₓ replaces the x-column with constants.
• D ≠ 0 gives a unique solution; D = 0 with one of Dₓ, D_y, D_z ≠ 0 means no solution; all zero means infinite solutions.

Example: Find |A| for $A=\left(\begin{array}{cc}2& 3\\ 5& 7\end{array}\right)$, and state whether A is invertible.

Solution: $|A|=(2)(7)-(3)(5)=14-15=-1\ne 0$, so A is non-singular and invertible — the inverse is $A^{-1}=\frac{1}{-1}\left(\begin{array}{cc}7& -3\\ -5& 2\end{array}\right)=\left(\begin{array}{cc}-7& 3\\ 5& -2\end{array}\right)$.

Example: Using a property, show that $\left|\begin{array}{ccc}1& a& b+c\\ 1& b& c+a\\ 1& c& a+b\end{array}\right|=0$.

Solution: Add columns 2 and 3 to column 3. Each entry of column 3 becomes a + b + c, so column 3 becomes a constant multiple of column 1. A determinant with two proportional columns is zero.

Exercises (NCERT Q1–Q59)

• 1) Evaluate the determinant
• 2) Evaluate the determinant
• 3) Evaluate the determinant
• 4) Evaluate the determinant
• 5) Evaluate the determinant
• 6) Evaluate the determinant
• 7) If , verify that
• 8) Evaluate the determinant
• 9) Find the values of x for which the determinants are equal:
• 10) If , find the value of x.
• 11) Using properties of determinants, show that
• 12) Using properties, show that
• 13) Using properties, evaluate
• 14) Using properties, show that
• 15) Using properties, show that
• 16) Using properties, show that
• 17) Using properties, show that
• 18) Using properties, show that
• 19) Using properties, show that
• 20) Using properties, show that
• 21) Using properties, show that
• 22) Using properties, show that
• 23) Using properties, show that
• 24) Using properties, show that
• 25) Using properties, show that
• 26) Using properties, show that
• 27) Using properties, show that
• 28) If a, b, c are in AP, then the determinant equals (A) 0, (B) 1, (C) x, (D) 2x.
• 29) Find the minor and cofactor of each element of the determinant
• 30) Find the minor and cofactor of the element in the determinant
• 31) Using cofactors of elements of , find |A|.
• 32) Find the cofactors of the elements of the determinant
• 33) Using cofactor expansion, evaluate
• 34) Find the minor and cofactor of each element of the determinant
• 35) Verify that the sum of the products of the elements of row 2 with their corresponding cofactors equals the determinant, for
• 36) Verify that for
• 37) Find the adjoint of the matrix
• 38) Find the inverse of using the adjoint method.
• 39) Find the inverse of using the adjoint method.
• 40) Verify that for and
• 41) Find the adjoint of the matrix
• 42) Verify that for
• 43) If A is a nonsingular matrix of order 3, verify that
• 44) Find the area of the triangle with vertices (1, 2), (4, 2) and (3, 5).
• 45) Find the area of the triangle with vertices (0, 0), (3, 4) and (-2, 3).
• 46) Find the area of the triangle with vertices (-1, -1), (4, -2) and (2, 2).
• 47) Find the area of the triangle formed by the points A(5, 7), B(3, 5) and C(3, 2).
• 48) Find the values of k for which the system of equations has a unique solution, using the determinant method.
• 49) Examine the consistency of the system
• 50) Solve the system using Cramer's rule:
• 51) Show that the points (1, -1), (-4, 2) and (-2, -3) are NOT collinear.
• 52) Examine the consistency of the system
• 53) Solve using Cramer's rule:
• 54) Find the area of the triangle formed by the line with the coordinate axes.
• 55) Find the value of k if the area of the triangle with vertices (k, 0), (4, 0) and (0, 2) is 4 square units.
• 56) Find the area of the triangle with vertices (2, -3), (4, 5) and (0, 2).
• 57) Examine the consistency of the system
• 58) Examine the consistency of the system
• 59) Examine the consistency of the system

Answers at a glance

1. 1
   Q1 → 18.
2. 2
   Q2 → 1.
3. 3
   Q3 → answer not in source — verify against the textbook
4. 4
   Q4 → 49.
5. 5
   Q5 → -5.
6. 6
   Q6 → 0.
7. 7
   Q7 → Verified: |2A| = -8 = 4|A|.
8. 8
   Q8 → abc - h^2(a + b + c) + 2h^3.
9. 9
   Q9 → answer not in source — verify against the textbook
10. 10
    Q10 → answer not in source — verify against the textbook
11. 11
    Q11 → Shown.
12. 12
    Q12 → Shown.
13. 13
    Q13 → -24.
14. 14
    Q14 → Shown.
15. 15
    Q15 → Shown.
16. 16
    Q16 → Shown.
17. 17
    Q17 → Shown.
18. 18
    Q18 → -1 is not 0, so the determinant is nonzero.
19. 19
    Q19 → Shown.
20. 20
    Q20 → answer not in source — verify against the textbook
21. 21
    Q21 → Shown: determinant = 0.
22. 22
    Q22 → Shown: determinant = 0.
23. 23
    Q23 → Shown: determinant = 0.
24. 24
    Q24 → Shown.
25. 25
    Q25 → Shown.
26. 26
    Q26 → answer not in source — verify against the textbook
27. 27
    Q27 → Shown: determinant = 0.
28. 28
    Q28 → Option (A) — 0.
29. 29
    Q29 → answer not in source — verify against the textbook
30. 30
    Q30 → M22 = 11, A22 = 11.
31. 31
    Q31 → |A| = 5.
32. 32
    Q32 → answer not in source — verify against the textbook
33. 33
    Q33 → -49.
34. 34
    Q34 → answer not in source — verify against the textbook
35. 35
    Q35 → Verified: both sums give 18.
36. 36
    Q36 → Verified: A(adj A) = (adj A)A = 0I = |A|I.
37. 37
    Q37 → adj(A) = [-5, -2; -3, 1].
38. 38
    Q38 → answer not in source — verify against the textbook
39. 39
    Q39 → answer not in source — verify against the textbook
40. 40
    Q40 → Verified.
41. 41
    Q41 → answer not in source — verify against the textbook
42. 42
    Q42 → Verified: A(adj A) = (adj A)A = -11I = |A|I.
43. 43
    Q43 → answer not in source — verify against the textbook
44. 44
    Q44 → 9/2 square units.
45. 45
    Q45 → 17/2 square units.
46. 46
    Q46 → 9 square units.
47. 47
    Q47 → 3 square units.
48. 48
    Q48 → Unique solution: x = 0, y = 1.
49. 49
    Q49 → Consistent; x = 3, y = 1.
50. 50
    Q50 → answer not in source — verify against the textbook
51. 51
    Q51 → The determinant is 19/2 which is not 0, so the points are not collinear.
52. 52
    Q52 → answer not in source — verify against the textbook
53. 53
    Q53 → answer not in source — verify against the textbook
54. 54
    Q54 → 10 square units.
55. 55
    Q55 → k = 0 or k = 8.
56. 56
    Q56 → 13 square units.
57. 57
    Q57 → Consistent; x = 2, y = -1.
58. 58
    Q58 → Inconsistent (no solution).
59. 59
    Q59 → Consistent with infinitely many solutions.

MCQ Questions

1 Mark Questions

1. 1
   Write the value of |1 2; 3 4|. (Ans: 1×4 − 2×3 = −2)
2. 2
   When is a square matrix said to be singular?
3. 3
   State the relation between A⁻¹ and adj A in terms of |A|. (Ans: A⁻¹ = adj A/|A|)
4. 4
   What is the cofactor of the element in row 2, column 1 of the matrix [1 2; 3 4]? (Ans: −3)

2 Mark Questions

1. 1
   If |2 4; 5 x| = 0, find x. (Ans: x = 10)
2. 2
   Evaluate the determinant |1 2 3; 0 4 5; 0 0 6| using its triangular form. (Ans: 1 × 4 × 6 = 24)
3. 3
   For A = [2 3; 1 4], verify A(adj A) = |A|I. (Ans: |A| = 5)

3 Mark Questions

1. 1
   Using properties of determinants, show that |a b c; a b² c²; bc ca ab| = (a − b)(b − c)(c − a)(ab + bc + ca).
2. 2
   Solve the equation |x 2 3; 1 x 1; 3 2 x| = 0 for x. (Ans: x = 4, 1, −4)
3. 3
   Find the adjoint of A = [1 2; 3 4] and use it to find A⁻¹. (Ans: adj A = [4 −2; −3 1], A⁻¹ = (1/−2) adj A)

4 Mark Questions

1. 1
   Examine whether the system x + 2y = 2, 2x + 4y = 8 is consistent by evaluating |A| and the ranks, and interpret the outcome. (Ans: |A| = 0, inconsistent — parallel lines)
2. 2
   Solve the system 2x − y = 3, 3x + 4y = 10 using the matrix method: write AX = B, find adj A and A⁻¹, and obtain x = 2, y = 1.