You could use sum to sum the elements of a matrix/array.
Question: why do you need the inverse? Is it to solve simultaneous equations? If it is, it would be faster to use Choleski LU decomposition than matrix inversion.
Reading chapter 2, you can judge for yourself about the best method and code it (the given codes are short and handy and don't need any library, I used to translate that given codes to F90 when needed).
P.S.: As far as I remember, Cholesky is a restrictive form of LU for symmetric matrices only
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