My guess is : Given $A$ a square matrix, then $A^{-1}=det(A)^{-1}adj(A)$ where $det(A)^{-1}$ is multiplication inverse of $det(A)$
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Your guess is correct: this algorithm works in any ring (whenever $\det A$ is invertible). – Crostul Apr 06 '17 at 07:12
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As spelled out in Hurkyl's answer (+1) the methods from a first course in linear algebra apply to any field. For a walkthru example of calculating the inverse of a 3x3 matrix over the field of 29 elements using elementary row operations see an old answer by yours truly. See also this. – Jyrki Lahtonen Apr 06 '17 at 07:19
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Most likely, whatever algorithms you would use to find the inverse of an invertible real matrix will work completely unchanged to find the inverse of an invertible matrix over any field. And the reason why they work will be similarly unchanged.
(I say "most likely" because there do exist some algorithms for doing matrix operations that depend somewhat on the field... but it's very unlikely that you would consider using them)
The formula you wrote is true, but its mainly meant for theoretical reasoning, not for doing computation.