f: GL_{n}(R)-->GL_{n}(R) is defined by f(A)= (A^{-1})^{T }. Show that f is not an

inner automorphism.

My Solution:

Assume that there is C in GL_{n}(R) such that CAC^{-1}= (A^{-1})^{T}for all A in GL_{n}(R).

Then for all A in GL_{n}(R)

det(CAC^{-1}) = det (A^{-1})^{T}

=> det(C). det(A). Det(C^{-1}) = det(A^{-1})

=> det(A) = det(A^{-1}).

But, the above is not true for all A in GL_{n}(R).

Therefore f is not an inner automorphism of GL_{n}(R).

Is this correct?

Please help!