True or following questions A and 13 are n Times n complex m

True or following questions, A and 13 are n Times n complex matrices. If A, B are Hermitian, AB must also be. If A is Hermitian and B is unitary, then BAB^-1 must be Hermitian. If A, B are unitary, AB must be unitary also. Det S = det(S*). If lambda is an eigenvalue of S, then lambda is always an eigenvalue of S*. If u(x, y) has continuous partial derivatives, there is always another function v(x, y) such that f(x + yi) = u(x, y) + iv(x, y) is complex differentiable. If lambda is Hermitian, then ||e^iA v|| = ||v|| for all v epsilon C^n.

Solution

i) Need not be true always

If A is real and B is real then A and B are symmetric and in that case only true.

ii) True since BAB^-1 = A and hence true

iii) A, B are unitary then AB is also unitary. true.

iv) True

As for any matrix det A = det A^tsince determinant is always real for Hermitian matrices

v) True as S* is formed by conjugates of S

vi) True

vii) True