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)`.

Explanation

`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`