If `A=[[cos alpha, -sin alpha] , [sin alpha, cos alpha]], B=[[cos2beta, sin 2beta] , [sin 2 beta, -cos2beta]]` where `0 lt beta lt pi/2` then prove that `BAB=A^(-1)` Also find the least positive value of `alpha` for which `BA^4B= A^(-1)`A.B.C.D.

If `A=[[cos alpha, -sin alpha] , [sin alpha, cos alpha]], B=[[cos2beta

1.	If `A=[[cos alpha, -sin alpha] , [sin alpha, cos alpha]], B=[[cos2beta, sin 2beta] , [sin 2 beta, -cos2beta]]` where `0 lt beta lt pi/2` then prove that `BAB=A^(-1)` Also find the least positive value of `alpha` for which `BA^4B= A^(-1)`A.B.C.D.
Answer» Correct Answer - `alpha =(2pi)/3` `because BAB=A^(-1)` `rArr ABAB= I` `rArr (AB)^(2) = I` Now, `AB= [[cos (alpha+2beta),sin (alpha+2beta)],[sin(alpha+2beta),-cos(alpha + 2beta)]]` and `(AB)^(2) = (AB) (AB) = [[1,0],[0,1]]= I [because (AB)(AB)=I]` Also, `BA^(4)B=A^(-1)` or `A^(4) B= B^(-1) A^(-1) =(AB)^(-1) = AB` or`A^(4) = A " "...(i)` Now, `A^(2) = [[cos alpha,-sin alpha],[sin alpha,cos alpha]][[cos alpha,-sin alpha],[sin alpha,cos alpha]]` `=[[cos 2alpha,-sin 2alpha],[sin 2alpha,cos 2alpha]]` Similarly, `A^(4)=[[cos 4alpha,-sin 4alpha],[sin 4alpha,cos4alpha]]` Hence, from Eq. (i) `[[cos 4alpha,-sin 4alpha],[sin 4alpha,cos4alpha]]=[[cos alpha,-sin alpha],[sin alpha,cos alpha]]` or `4 alpha = 2pi + alpha` ` therefore alpha = (2pi)/3`

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If `A=[[cos alpha, -sin alpha] , [sin alpha, cos alpha]], B=[[cos2beta, sin 2beta] , [sin 2 beta, -cos2beta]]` where `0 lt beta lt pi/2` then prove that `BAB=A^(-1)` Also find the least positive value of `alpha` for which `BA^4B= A^(-1)`A.B.C.D.

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