## How to caculate the inverse metric tensor

Given a metric tensor gmn, how to calculate the inverse of it, gmn. For example, the metric
$$g_{\mu \nu }= \left[ \begin{array}{cccc} f & 0 & 0 & -w \\ 0 & -e^m & 0 &0 \\0 & 0 & -e^m &0\\0 & 0 & 0 & -l \end{array} \right]$$

From basic understanding, I would think of divided it, that is
$$g^{\mu \nu }= \left[ \begin{array}{cccc} 1/f & 0 & 0 & -1/w \\ 0 & -e^{-m} & 0 &0 \\0 & 0 & -e^{-m} &0\\0 & 0 & 0 & -1/l \end{array} \right]$$

But the author gave some different answer, that is
$$g^{\mu \nu }= \left[ \begin{array}{cccc} \frac{l}{fl+w^2} & 0 & 0 & -\frac{w}{fl+w^2} \\ 0 & -e^{-m} & 0 &0 \\0 & 0 & -e^{-m} &0\\0 & 0 & 0 & -\frac{f}{fl+w^2}\end{array} \right]$$

So how should I calculate the inverse metric tensor as shown by the author?

Thanks

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 The inverse metric is, like the name suggests, just the inverse matrix. You should have learned how to calculate inverse matrices in Linear algebra, there are many methods. The way you are trying, where you just invert the entries only works if the matrix is diagonal, which this one is not.

 Quote by Matterwave The inverse metric is, like the name suggests, just the inverse matrix. You should have learned how to calculate inverse matrices in Linear algebra, there are many methods. The way you are trying, where you just invert the entries only works if the matrix is diagonal, which this one is not.
Well, I thought the same way at first but I have tried find the inverse matrix but the result is different from the given one.
Mathematica give the me the inverse matrix as
$$g^{\mu \nu }= \left[ \begin{array}{cccc} \frac{1}{f} & 0 & 0 & -\frac{w}{fl} \\ 0 & -e^{-m} & 0 &0 \\0 & 0 & -e^{-m} &0\\0 & 0 & 0 & -\frac{1}{l} \end{array} \right]$$

That's why I wonder how the author get that result..

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## How to caculate the inverse metric tensor

Well in the first place you have written the metric tensor incorrectly. It should be symmetric. Also you have either entered it into Mathematica incorrectly, or incorrectly copied down the result. Really, you should be able to invert a simple matrix like this all by yourself!

 Quote by Bill_K Well in the first place you have written the metric tensor incorrectly. It should be symmetric. Also you have either entered it into Mathematica incorrectly, or incorrectly copied down the result. Really, you should be able to invert a simple matrix like this all by yourself!
Nope. It is correct. I copied the metric tensor and its inverse directly from the Mallett paper. I checked it many times. I copied it correctly. It is from Mallett paper on circulating beam. I dont think it is written wrongly in the paper.

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