Yes, your proof is correct.
You could also provide a counterexample. For instance, let and . Then . Thus for any whereas .
f: GLn(R)-->GLn(R) is defined by f(A)= (A-1)T . Show that f is not an
Assume that there is C in GLn(R) such that CAC-1 = (A-1)T for all A in GLn(R).
Then for all A in GLn(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 GLn(R).
Therefore f is not an inner automorphism of GLn(R).
Is this correct?