If `P= [[sqrt(3)/2, 1/2],[-1/2 , sqrt(3)/ 2]], A = [[1,1],[0,1]]and Q= PAP^(T)` , the ltbr. `P^(T)(Q^(2005)) P` equal to

Options

A. `[[1,2005],[0,1]]`
B. `[[sqrt(3)//1,2005],[1,0]]`
C. `[[1, 2005],[sqrt(3)//2,1]]`
D. `[[1, sqrt(3)//2 ],[0, 2005]]`

Answer

A

Explanation

If `Q = PAP^(T) `
then `P^(T)Q=AP^(T) " "[because PP^(T)=I]`
` rArrP^(T) Q^(2005) P = AP^(T)Q^(2004)P`
`= A^(2) P^(T) Q ^(2003) P = A^(3) P^(T) Q^(2002) P `
`= A^(2004)P^(T)(QP)`
`= A^(2004)P^(T)(PA)" "[Q= PAP^(T) rArr QP= PA]`
`=A^(2005)`
`therefore A^(2005) = [[1,2005],[0,1]]`