Maths-
General
Easy

Question

If w is a complex cube root of unity, then the matrix A = open square brackets table row 1 cell w to the power of 2 end exponent end cell w row cell w to the power of 2 end exponent end cell w 1 row w 1 cell w to the power of 2 end exponent end cell end table close square brackets is a-

  1. singular matrix    
  2. non-singular matrix    
  3. skew symmetric matrix    
  4. None of these    

The correct answer is: singular matrix


    We have

    |A|=open square brackets table row 1 cell w to the power of 2 end exponent end cell w row cell w to the power of 2 end exponent end cell w 1 row w 1 cell w to the power of 2 end exponent end cell end table close square brackets=open square brackets table row cell 1 plus w to the power of 2 end exponent plus w end cell cell w to the power of 2 end exponent end cell w row cell w to the power of 2 end exponent plus w plus 1 end cell w 1 row cell w plus 1 plus w to the power of 2 end exponent end cell 1 cell w to the power of 2 end exponent end cell end table close square brackets [Using C subscript 1 end subscript rightwards arrow C subscript 1 end subscript plus C subscript 2 end subscript plus C subscript 3 end subscript]

    = open square brackets table row 0 cell w to the power of 2 end exponent end cell w row 0 w 1 row 0 1 cell w to the power of 2 end exponent end cell end table close square brackets = 0

    A is a singular matrix.

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