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Exercise - 4.5

Determinants

**Question-1 :-**
Find adjoint of each of the matrices:

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

**Question-2 :-**
Find adjoint of each of the matrices:

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

**Question-3 :-**
Verify A (adj A) = (adj A) A = |A| I

|A| = -12 + 12 = 0 Minor of the element aᵢⱼ is Mᵢⱼ Here a₁₁ = 1. So M₁₁ = Minor of the element a₁₁= -6 M₁₂ = Minor of the element a₁₂ = -4 M₂₁ = Minor of the element a₂₁ = 3 M₂₂ = Minor of the element a₂₂ = 2 Now, cofactor of aᵢⱼ is Aᵢⱼ. So A₁₁ = (–1) ¹⁺¹ M₁₁ = (–1)^{2}(-6) = -6 A₁₂ = (–1) ¹⁺² M₁₂ = (–1)^{3}(-4) = 4 A₂₁ = (–1) ²⁺¹ M₂₁ = (–1)^{3}(3) = -3 A₂₂ = (–1) ²⁺² M₂₂ = (–1)^{4}(2) = 2

**Question-4 :-**
Verify A (adj A) = (adj A) A = |A| I

|A| = 1(0 - 0) + 1(9 + 2) + 2(0 - 0) = 0 + 11 + 0 = 11 Minors: - Cofactors: - M₁₁ = 0 – 0 = 0 A₁₁ = (–1) ¹⁺¹ (0) = 0 M₁₂ = 9 + 2 = 11 A₁₂ = (–1) ¹⁺² (11) = -11 M₁₃ = 0 - 0 = 0 A₁₃ = (–1) ¹⁺³ (0) = 0 M₂₁ = -3 – 0 = -3 A₂₁ = (–1) ²⁺¹ (-3) = 3 M₂₂ = 3 - 2 = 1 A₂₂ = (–1) ²⁺² (1) = 1 M₂₃ = 0 + 1 = 1 A₂₃ = (–1) ²⁺³ (1) = -1 M₃₁ = 2 - 0 = 2 A₃₁ = (–1) ³⁺¹ (2) = 2 M₃₂ = -2 – 6 = -8 A₃₂ = (–1) ³⁺² (-8) = 8 M₃₃ = 0 + 3 = 3 A₃₃ = (–1) ³⁺³ (3) = 3

**Question-5 :-**
Find the inverse of each of the matrices (if it exists) given below:

**Question-6 :-**
Find the inverse of each of the matrices (if it exists) given below:

**Question-7 :-**
Find the inverse of each of the matrices (if it exists) given below:

**Question-8 :-**
Find the inverse of each of the matrices (if it exists) given below:

**Question-9 :-**
Find the inverse of each of the matrices (if it exists) given below:

**Question-10 :-**
Find the inverse of each of the matrices (if it exists) given below:

**Question-11 :-**
Find the inverse of each of the matrices (if it exists) given below:

**Question-12 :-**

**Question-13 :-**

**Question-14 :-**

**Question-15 :-**

**Question-16 :-**

**Question-17 :-**
Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to

(A) |A| (B) |A|^{2} (C) |A|^{3} (D) 3|A|

The correct answer is B.

**Question-18 :-**
If A is an invertible matrix of order 2, then det (A^{-1}) is equal to

(A) det(A) (B) 1/det(A) (C) 1 (D) 0

The correct answer is B.

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