Exercise 1.07 Tensors and Vectors

The Question was

To see my answer Click Here. The formulae are too complicated to put into this simple editor.

One thing that Sean Carroll had omitted to tell us was that an index can be raised or lowered by applying the metric tensor. So for example, 

X^μ_ν = η_νσ X^σμ

This is pretty vital for this exercise! The Wikipedia article is here.

X_[μν] = X^[μν]

I don't know if that is significant.