If `A A^T=I` and C is a skew-symmetric matrix then `[(A^TCA)^50]^T` equals (A) `A^50(C^T)^50(A^T)^50` (B) `A^TC^50A` (C) `-A^TC^50A` (D) `-AC^50A^T`

Explanation

As `A A^T = I =>(A A^T)^n = (A A^T)`

`[(A^TCA)^50]^T = [(A^T)^50C^50A^50]^T`

As `[AB]^T = B^TA^T`, our expression becomes,

`=(A^50)^T(C^50)^T((A^T)^50)^T`

`=(A^T)^50(C^T)^50(A^50)` ..... (As `(A^T)^T = A`)

`=(A^T)^50(-C)^50(A^50) `......(As `C^T = -C`)

`=A^TC^50A`

So,option `B` is the correct option.