TOPICS

Exercise - 4.4

Determinants

**Question-1 :-**
Write Minors and Cofactors of the elements of following determinants:

(i) Minor of the element aᵢⱼ is Mᵢⱼ Here a₁₁ = 1. So M₁₁ = Minor of the element a₁₁= 3 M₁₂ = Minor of the element a₁₂ = 0 M₂₁ = Minor of the element a₂₁ = –4 M₂₂ = Minor of the element a₂₂ = 2 Now, cofactor of aᵢⱼ is Aᵢⱼ. So A₁₁ = (–1) ¹⁺¹ M₁₁ = (–1)^{2}(3) = 3 A₁₂ = (–1) ¹⁺² M₁₂ = (–1)^{3}(0) = 0 A₂₁ = (–1) ²⁺¹ M₂₁ = (–1)^{3}(–4) = 4 A₂₂ = (–1) ²⁺² M₂₂ = (–1)^{4}(2) = 2

(ii) Minor of the element aᵢⱼ is Mᵢⱼ Here a₁₁ = 1. So M₁₁ = Minor of the element a₁₁= d M₁₂ = Minor of the element a₁₂ = b M₂₁ = Minor of the element a₂₁ = c M₂₂ = Minor of the element a₂₂ = a Now, cofactor of aᵢⱼ is Aᵢⱼ. So A₁₁ = (–1) ¹⁺¹ M₁₁ = (–1)^{2}(d) = d A₁₂ = (–1) ¹⁺² M₁₂ = (–1)^{3}(b) = -b A₂₁ = (–1) ²⁺¹ M₂₁ = (–1)^{3}(c) = -c A₂₂ = (–1) ²⁺² M₂₂ = (–1)^{4}(a) = a

**Question-2 :-**
Write Minors and Cofactors of the elements of following determinants:

(i) Minors: - Cofactors: - M₁₁ = 1 – 0 = 1 A₁₁ = (–1) ¹⁺¹ (1) = 1 M₁₂ = 0 - 0 = 0 A₁₂ = (–1) ¹⁺² (0) = 0 M₁₃ = 0 – 0 = 0 A₁₃ = (–1) ¹⁺³ (0) = 0 M₂₁ = 0 – 0 = 0 A₂₁ = (–1) ²⁺¹ (0) = 0 M₂₂ = 1 – 0 = 1 A₂₂ = (–1) ²⁺² (1) = 1 M₂₃ = 0 - 0 = 0 A₂₃ = (–1) ²⁺³ (0) = 0 M₃₁ = 0 - 0 = 0 A₃₁ = (–1) ³⁺¹ (0) = 0 M₃₂ = 0 – 0 = 0 A₃₂ = (–1) ³⁺² (0) = 0 M₃₃ = 1 - 0 = 1 A₃₃ = (–1) ³⁺³ (1) = 1

(ii) Minors: - Cofactors: - M₁₁ = 10 + 1 = 11 A₁₁ = (–1) ¹⁺¹ (11) = 11 M₁₂ = 6 - 0 = 6 A₁₂ = (–1) ¹⁺² (6) = -6 M₁₃ = 3 – 0 = 3 A₁₃ = (–1) ¹⁺³ (3) = 3 M₂₁ = 0 – 4 = -4 A₂₁ = (–1) ²⁺¹ (-4) = 4 M₂₂ = 2 – 0 = 2 A₂₂ = (–1) ²⁺² (2) = 2 M₂₃ = 1 - 0 = 1 A₂₃ = (–1) ²⁺³ (1) = -1 M₃₁ = 0 - 20 = -20 A₃₁ = (–1) ³⁺¹ (-20) = -20 M₃₂ = -1 – 12 = -13 A₃₂ = (–1) ³⁺² (-13) = 13 M₃₃ = 5 - 0 = 5 A₃₃ = (–1) ³⁺³ (5) = 5

**Question-3 :-**
Using Cofactors of elements of second row, evaluate

Elements are : a₂₁ = 2, a₂₂ = 0, a₂₃ = 1 Minors: - Cofactors: - M₂₁ = 9 – 16 = -7 A₂₁ = (–1) ²⁺¹ (-7) = 7 M₂₂ = 15 – 8 = 7 A₂₂ = (–1) ²⁺² (7) = 7 M₂₃ = 10 - 3 = 7 A₂₃ = (–1) ²⁺³ (7) = -7 a₂₁ A₂₁ + a₂₂ A₂₂ + a₂₃ A₂₃ = 2 x 7 + 0 x 7 + 1 x (-7) = 14 + 0 - 7 = 7

**Question-4 :-**
Using Cofactors of elements of third column, evaluate

Elements are : a₁₃ = yz, a₂₃ = zx, a₃₃ = xy Minors: - Cofactors: - M₁₃ = z – y A₁₃ = (–1) ¹⁺³ (z – y) = z – y M₂₃ = z – x A₂₃ = (–1) ²⁺³ (z – x) = x - z M₃₃ = y - x A₃₃ = (–1) ³⁺³ (y - x) = y - x a₁₃ A₁₃ + a₂₃ A₂₃ + a₃₃ A₃₃ = yz . (z - y) + zx . (x - z) + xy . (y - x) = yz² - y²z + zx² - z²x + xy² - x²y = (x²z - y²z) + (yz² - xz²) + (xy² - yx²) = z(x² - y²) + z²(y - x) + xy(y - x) = z(x + y)(x - y) - z²(x - y) - xy(x - y) = (x - y)[zx + zy - z² - xy] = (x - y)[z(x - z) + y(z - x)] = (x - y)(x - z)(y - z)

**Question-5 :-**
and Aᵢⱼ is Cofactors of aᵢⱼ, then value of Δ is given by

(A) a₁₁ A₃₁ + a₁₂ A₃₂ + a₁₃ A₃₃

(B) a₁₁ A₁₁ + a₁₂ A₂₁ + a₁₃ A₃₁

(C) a₂₁ A₁₁ + a₂₂ A₁₂ + a₂₃ A₁₃

(D) a₁₁ A₁₁ + a₂₁ A₂₁ + a₃₁ A₃₁

a₁₁ A₁₁ + a₂₁ A₂₁ + a₃₁ A₃₁ The correct answer is D.

CLASSES