Maths-
General
Easy

Question

Let omega not equal to 1 be a cube root of unity and S be the set of all non‐singular matrices of the form open square brackets table row 1 a b row omega 1 c row cell omega to the power of 2 end exponent end cell omega 1 end table close square brackets, where each of a, b and c is either omega text  or  end text omega to the power of 2 end exponent Then the number of distinct matrices in the set S is‐

  1. 2    
  2. 6    
  3. 4    
  4. 8    

The correct answer is: 2


    Given questionLet omega not equal to 1 be a cube root of unity and S be the set of all non‐singular matrices of the form of matrix
    Given Matrix
    open square brackets table row 1 a b row omega 1 c row cell omega squared end cell omega 1 end table close square brackets
    Determinant D is not equal to 0
    1 left parenthesis 1 minus omega c right parenthesis minus a left parenthesis omega minus omega squared c right parenthesis plus b left parenthesis omega squared space space minus omega squared right parenthesis not equal to 0
1 minus omega left parenthesis a plus c right parenthesis plus a c omega squared space not equal to 0
a plus c not equal to negative 1
a c not equal to 1
S o comma space a equals omega space o r space omega squared space space space a n d space c equals omega space o r space omega squared space space space
space I f space c equals omega squared space comma space t h e n space capital delta equals 0. space S o space c not equal to omega squared space space space S o comma space c equals omega space space space
S o comma space b y space e q u a t i o n space left parenthesis 1 right parenthesis comma space a not equal to omega squared space space. space space
H e n c e comma space space a equals omega. space space
S i n c e comma space t h e space d e t e r m i n a n t space v a l u e space i s space i n d e p e n d e n t space o f space b space. space S o space b space c a n space b e space omega space o r space omega squared space space. space space
H e n c e comma space c equals omega comma a equals omega comma b equals omega space o r space omega squared space space. space space
S o comma space n u m b e r space o f space m a t r i c e s space f o r m e d space w i l l space b e space 2.

    Hence no of matrices formed = 2

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