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If A = [{:("cos" theta, -"sin"theta),("s...

If `A = [{:("cos" theta, -"sin"theta),("sin"theta, "cos"theta):}]`, then the matrix `A^(-50) " when " theta = (pi)/(12)`, is equal to

A

`[(1/2, (sqrt(3))/2),(-(sqrt(3))/2,1/2)]`

B

`[((sqrt(3))/2,-1/2),(1/2,(sqrt(3))/2)]`

C

`[(1/2,(sqrt(3))/2),(1/2,1/2)]`

D

`[((sqrt(3))/2,1/2),(-1/2,(sqrt(3))/2)]`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `A (theta)=[(cos theta, -sin theta),(sin theta, cos theta)]`
Then `A(theta0xxA(alpha)=A(alpha+theta)`
Also `A(0)=I`
Hence`[A(theta)]^(-1)=A(-theta` and `[A(theta)]^(n)=A( n theta) AA n epsilon ZimpliesA^(-50)(theta)=A(-50 theta)`
`theta=(pi)/12 impliesA^(-50)((pi)/12)=A(-25(pi)/6)=A((-pi)/6)=[(sqrt(3)//2,1//2),(-1//2,sqrt(3)//2)]`
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