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Exercise - 4.5

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

Solution :-
  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
  determinant
    

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

Solution :-
  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
  determinant
    

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

Solution :-
  |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
  determinant
    

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

Solution :-
  |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
determinant
    

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

Solution :-
 determinant
    

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

Solution :-
determinant
    

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

Solution :-
determinant
    

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

Solution :-
determinant
    

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

Solution :-
determinant
    

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

Solution :-
determinant
    

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

Solution :-
    determinant
    

Question-12 :-  determinant

Solution :-
determinant
    

Question-13 :-  determinant

Solution :-
determinant
    

Question-14 :-  determinant

Solution :-
    determinant
    

Question-15 :-  determinant

Solution :-
determinant
determinant
    

Question-16 :-  determinant

Solution :-
determinant
determinant
    

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|

Solution :-
determinant
  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

Solution :-
determinant
  The correct answer is B.
    
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