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KVPY2010MathematicsMatrices

Let A denote the matrix ( array ll 0 & i i & 0 array ) , where i²=-1 , and let I denote the identity matrix ( array ll 1 & 0 0 & 1 array ) . Then I + A + A ²+ + A ²⁰¹⁰ is -

Options

  1. A( array ll 0 & 0 0 & 0 array )
  2. B( array ll 0 & i i & 0 array )
  3. C( array ll 1 & 1 i & 1 array )
  4. D( array cc -1 & 0 0 & -1 array )

Correct answer

C. ( array ll 1 & 1 i & 1 array )

Step-by-step solution

array l A = [ array ll 0 & i i & 0 array ] ; A ²= [ array cc -1 & 0 0 & -1 array ] ; A ³= [ array cc 0 & - i - i & 0 array ] ; I = [ array ll 1 & 0 0 & 1 array ] I + A + A ²+ A ³= [ array ll 0 & 0 0 & 0 array ] ; A ⁴= [ array ll 1 & 0 0 & 1 array ]= I I + A + A ²+ A ³+ A ²¹⁰ ( I + A + A ²+ A ³ )+ A ⁴ ( I + A + A ²+ A ³ )+ + A ²⁰⁰⁸ ( I + A + A ² ) = ( array ll 1 & i i & 1 array ) array

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