Matrix Index Inversion: Clarification Needed

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is it true that \frac{1}{g_{ab}}=g^{ba}? I am a bit confused by the index notation. I especially wonder about the inversion of the indices. Could somebody clarify this please?
 
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No, that's not true. That would be the matrix with reciprocal entries, which is obviously not the inverse.

It would take me a while to explain the index notation and lowering and raising indices (and some Latex work), which I am not feeling up to right now.
 
##g^{ab}## is the number on row a, column b of the inverse of the matrix that has ##g_{ab}## on row a, column b.

It's not true in general that if A is an invertible matrix, then ##(A^{-1})_{ij}=1/A_{ji}##. Even when A is diagonal, it's only true for the numbers on the diagonal.
 
IF g^{ij} is intended as the fundamental metric tensor, ds^2= g^{ij}dx_idx_j, then it is true that g_{ij}= (g^{ij})^{-1} but, again, that is NOT the same as \frac{1}{g_{ij}}.
 

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