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KCET2018MathematicsMatrices

If ( A= [ array cc 2 & -2 -2 & 2 array ] ), then ( A^ n =2^ k A ), where ( K= )

Options

  1. A( 2^ n-1 )
  2. Bn ( +1 )
  3. Cn-1
  4. D( 2(n-1) )

Correct answer

D. ( 2(n-1) )

Step-by-step solution

Given that ( A= [ array cc 2 & -2 -2 & 2 array ] ) and ( A^ n =2^ k A ) So, ( A²= [ array cc 2 & -2 -2 & 2 array ] [ array cc 2 & -2 -2 & 2 array ] ) ( A³= [ array cc 8 & -8 -8 & 8 array ]=2² A ) ( = [ array cc 32 & -32 -32 & 32 array ]=2⁴ A ) ( A⁴= [ array cc 32 & -32 -32 & 32 array ] [ array cc 2 & -2 -2 & 2 array ] ) ( = [ array cc 128 & -128 -128 & 128 array ]=2⁶ A ) So, ( k=2(n-1) )

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