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If {:P=[(sqrt3/2,1/2),(1/2,sqrt3/2)],A=[...

If `{:P=[(sqrt3/2,1/2),(1/2,sqrt3/2)],A=[(1,1),(0,1)]:}and Q=PAP^T," then" P^TQ^2015P`, is

A

`{:[(1,2005),(0,1)]:}`

B

`{:[(1,2005),(2005,1)]:}`

C

`{:[(1,0),(2005,1)]:}`

D

`{:[(1,0),(0,1)]:}`

Text Solution

Verified by Experts

The correct Answer is:
A
We have,
`P P^T={:[(sqrt3/2,1/2),(-1/2,sqrt3/2)][(sqrt3/2,-1/sqrt2),(1/2,sqrt3/2)]=[(1,0),(0,1)]=I:}`
`:. P^T=P^(-1)`
Now,
`Q=PAP^T`
`rArr P^TQP=P^T(PAP^T)P`
`rArr P^TQP=(P^TP)A(P^TP)=IAI=A={:[(1,1),(0,1)]:}`
`and, Q_2=(PAP^T)(PAP^T)=PA(P^TP)AP^T=PA^2P^T`
`:. P^TQ^2P=P^T(PA^2P^T)P=(P^TP)A^2(P^TP)=IA^2`
`rArr P^TQ^2P=A^2={:[(1,2),(0,1)]:}`
Continuing in this manner, we have
`P^TQ^2005P=A^2005=`{:[(1,2005),(0,1)]:}`
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