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Let a matrix A = open square brackets table row 1 1 row 0 1 end table close square brackets, P = open square brackets table row cell fraction numerator square root of 3 over denominator 2 end fraction end cell cell fraction numerator 1 over denominator 2 end fraction end cell row cell negative fraction numerator 1 over denominator 2 end fraction end cell cell fraction numerator square root of 3 over denominator 2 end fraction end cell end table close square brackets, Q = PAPT where PT is transpose of matrix P. then PT Q2005 P is-

  1. open square brackets table row 1 2005 row 0 1 end table close square brackets    
  2. fraction numerator 1 over denominator 4 end fraction open square brackets table row cell 1 plus 2005 square root of 3 end cell 6015 row 2005 cell 1 minus 2005 square root of 3 end cell end table close square brackets    
  3. fraction numerator 1 over denominator 4 end fraction open square brackets table row cell 1 plus 2005 square root of 3 end cell 2005 row 2005 cell 1 minus 2005 square root of 3 end cell end table close square brackets    
  4. open square brackets table row 2005 2005 row 0 1 end table close square brackets    

The correct answer is: open square brackets table row 1 2005 row 0 1 end table close square brackets


    PT Q2005 P = PT (PAPT)2005P
    = PT fraction numerator open curly brackets left parenthesis P A P to the power of T end exponent right parenthesis left parenthesis P A P to the power of T end exponent right parenthesis........ left parenthesis P A P to the power of T end exponent right parenthesis close curly brackets over denominator 2005 t i m e s end fractionP
    = fraction numerator left parenthesis P to the power of T end exponent P right parenthesis A left parenthesis P to the power of T end exponent P right parenthesis A left parenthesis P to the power of T end exponent P right parenthesis........ left parenthesis P to the power of T end exponent P right parenthesis A left parenthesis P to the power of T end exponent P right parenthesis over denominator 2005 t i m e s end fraction= A2005
    A2 =open square brackets table row 1 2 row 0 1 end table close square brackets, A3 = A2A = open square brackets table row 1 3 row 0 1 end table close square brackets…. and so on.
    A2005 = open square brackets table row 1 2005 row 0 1 end table close square bracketsrightwards double arrow PT Q2005 P = open square brackets table row 1 2005 row 0 1 end table close square brackets

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