If `P=[[sqrt(3)/2,1/2],[-1/2,sqrt(3)/2]], A=[[1,1],[0,1]]` and `Q=PAP^T` and `X=P^TQ^(2005)P`, then `X` equal to:
A. `[[1,2005],[0,1]]`
B. `[[4+2005sqrt3,6015],[2005,4-2005sqrt3]]`
C. `1/4 [[2+sqrt3,1],[-1,2-sqrt3]]`
D. `1/4 [[2005,2-sqrt3],[2+sqrt3,2005]]`

Explanation

`P = [[sqrt3/2,1/2],[-1/2, sqrt3/2]]`

`:. P^T = [[sqrt3/2,-1/2],[1/2, sqrt3/2]]`

Now, `P^TP = [[1,0],[0,1]] = I`

Now, `P^TQ^2005P = P^T(PAP^T)^2005P = P^T(PAP^T)(PAP^T)(PAP^T)...P = A^2005`

Here, `A = [[1,1],[0,1]]`

`A^2 = A.A = [[1,1],[0,1]][[1,1],[0,1]] = [[1,2],[0,1]]`

`A^3 = A^2.A = [[1,2],[0,1]][[1,1],[0,1]] = [[1,3],[0,1]]`

`:. A^2005 = [[1,2005],[0,1]]`

...