For each of the following, determine if the given operation defines an inner product on the given vector space V. If it does,

Question

For each of the following, determine if the given operation defines an inner product on the given vector space V. If it does, show that it does by verifying that axioms (i), (ii), (iii), (iv) of an inner product holds. If it does not, give a counterexample to show that one of the axioms (i), (ii), (iii), (iv) fails.
(a) V = M₂ (R), (A, B) = det(AB)
(b) V = M₂ (R), (A, B) = Tr(ATB)
(c) V = F, (f, g) = f(1)g(0) + f(0)g(1)

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