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If P=[[sqrt(3)/2,1/2],[-1/2,sqrt(3)/2]],...

If `P=[[sqrt(3)/2,1/2],[-1/2,sqrt(3)/2]], A=[[1,1],[0,1]]` and `Q=PAP^T`, then `P^TQ^(2015)P` is

Text Solution

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`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^2015P = P^T(PAP^T)^2015P = P^T(PAP^T)(PAP^T)(PAP^T)...P = A^2015`
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^2015 = [[1,2015],[0,1]]`
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