Do any of the following define homomorphisms from GL(n, C) to GL(n, C) for n ≥ 2?Give brief explanations.(a) f (A) = AT (b) f (A) = (A−1)T (c) f (A) = A2

(a) f(A) = AT

This does not define a homomorphism from GL(n, C) to GL(n, C). A homomorphism is a map that preserves the structure of the group, meaning that for any two elements A and B in GL(n, C), f(AB) = f(A)f(B). However, for the transpose operation, (AB)T = BT AT, which is not equal to AT BT = f(A)f(B). Therefore, f(A) = AT does not define a homomorphism.

(b) f(A) = (A−1)T

This does define a homomorphism from GL(n, C) to GL(n, C). For any two elements A and B in GL(n, C), f(AB) = (AB)−1T = (B−1 A−1)T = (A−1)T (B−1)T = f(A)f(B). Therefore, f(A) = (A−1)T does define a homomorphism.

(c) f(A) = A2

This does define a homomorphism from GL(n, C) to GL(n, C). For any two elements A and B in GL(n, C), f(AB) = (AB)2 = ABAB = A(BA)B = A2 B2 = f(A)f(B). Therefore, f(A) = A2 does define a homomorphism.