If `P = [[cos frac(pi)(6), sin frac(pi)(6) ],[-sinfrac(pi)(6),cosfrac(pi)(6)]], A = [[1,1],[0,1]]and Q = PAP^(T)` then
`P^(T) Q^(2007) P ` is equal to

Options

A. `[[1,sqrt(3)//2],[0,2007]]`
B. `[[1,2007],[0,1]]`
C. `[[sqrt(3)//2, 2007 ],[0,1]]`
D. `[[sqrt(3)//2,-1//2],[1,2007]]`

Answer

B

Explanation

We have, `P^(T) = P^(-1) " " [ because PP^(T) =I]`
Now, ,,`Q= PAP^(T) = PAP^(-1)`
`therefore Q^(2007) = PA^(2007) P^(-1) `
`therefore P^(T) Q^(2007 ) P= P^(-1) (PA^(2007)P^(-1)) P`
`= A^(2007 = [[1,2007],[0,1]][becauseA^(2)=[[1,2],[0,1]],A^(3) = [[1,3],[0,1]],...]`