Calculate the metric and conjugate metric components for oblique cylindrical coordinates

where:

,

,

and

is a parameter.

I can calculate the metric components but want to know how I would be expected to calculate the conjugate metric components.

For the metric components I get:

It seems to me that I need to either find

in terms of x,y,z and then calculate the conjugate basis or alternatively I need to find the inverse of the matrix above. Either task seems to be too hard for the number of marks awarded to the question.