If `A` is square matrix of order `2xx2` such that `A^(2)=I,B=[{:(1,sqrt(2)),(0,1):}]` and `C=ABA` then

Options

A. `C^(2009)=A[{:(1,2009-sqrt(2)),(0,1):}]`
B. `C^(2009)=A[{:(1,2009+sqrt(2)),(0,1):}]`
C. `|C|^(2009)=1`
D. `C^(2009)=A[{:(1,2009sqrt(2)),(0,1):}]`

Answer

C, D

Explanation

`(ABA)^(2)=(ABA)(ABA)=AB(AA)BA=AB^(2)A`
continuing in the manner `C^(2008)=AB^(2009)A`
`B^(2)=[{:(1,2sqrt(2)),(0,1):}],B^(3)=[{:(1,3sqrt(2)),(0,1):}]`
Continuing in this manner `B^(2009)=[{:(1,2009sqrt(2)),(0,1):}]`