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Question 29594: Determine whether the following is a group homomorphism.
x: GLn(R) to GLn(R), x(A)=A^T
T is the transpose
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! x: GLn(R) to GLn(R), x(A)=A^T
Note: x(A^-1) = (A^-1)^T= (A^T)^-1 = x(A)^-1
(so x preserves the inverse operation)
but x(AB) = (AB)^T = B^T A^T = x(B) x(A).
e.g. A = (0 0)
(1 0)
B= (0 -1)
(0 0).
AB = (0 0)
(0 -1).
x(A) = (0 1)
(0 0)
x(B)= (0 0)
(-1 0)
you can see x(AB)= (AB)^T =
(0 0)
(0 -1)
while x(A) x(B) =
(-1 0)
(0 0)
Since, in general, A & B don't commute in GLn(R),
so x cannot be a group homo.
Kenny
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