Let `A=[("tan"pi/3,"sec" (2pi)/3),(cot (2013 pi/3),cos (2012 pi))]` and P be a `2 xx 2` matrix such that `P P^(T)=I`, where I is an identity matrix of order 2. If `Q=PAP^(T)` and `R=[r_("ij")]_(2xx2)=P^(T) Q^(8) P`, then find `r_(11)`.

Let `A=[("tan"pi/3,"sec" (2pi)/3),(cot (2013 pi/3),cos (2012 pi))]` an

1.	Let `A=[("tan"pi/3,"sec" (2pi)/3),(cot (2013 pi/3),cos (2012 pi))]` and P be a `2 xx 2` matrix such that `P P^(T)=I`, where I is an identity matrix of order 2. If `Q=PAP^(T)` and `R=[r_("ij")]_(2xx2)=P^(T) Q^(8) P`, then find `r_(11)`.
Answer» `R=P^(T)Q^(8)P` `=P^(T)(PAP^(T))^(8)P` `=P^(T)PAP^(T) (PAP^(T))^(7)P` `=IAP^(T) (PAP^(T))^(7) P` `=AP^(T) PAP^(T) (PAP^(T))^(6)P` `=A^(2)P^(T) (PAP^(T))^(6)P` ... ... `=A^(8) P^(T)P` `=A^(8)` Now, `A^(2)=A A=[(sqrt(3),-2),(0,1)][(sqrt(3),-2),(0,1)]` `=[((sqrt(3))^(2),-2sqrt(3)-2),(0,1)]` `A^(3)=A^(2)A=[(3,-2sqrt(3)-2),(0,1)][(sqrt(3),-2),(0,1)]` `=[((sqrt(3))^(3),-6-2sqrt(3)-2),(0,1)]` `:. R=[r_("ij")]_(2xx2)=P^(T) Q^(8) P=A^(8)=[((sqrt(3))^(8),-),(-,-)]` `implies r_(11)=81`

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Let `A=[("tan"pi/3,"sec" (2pi)/3),(cot (2013 pi/3),cos (2012 pi))]` and P be a `2 xx 2` matrix such that `P P^(T)=I`, where I is an identity matrix of order 2. If `Q=PAP^(T)` and `R=[r_("ij")]_(2xx2)=P^(T) Q^(8) P`, then find `r_(11)`.

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