Yes, your proof is correct.
You could also provide a counterexample. For instance, let and . Then . Thus for any whereas .
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!