Matrix Index Inversion: Clarification Needed

[gentsagree]

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?

 

[homeomorphic]

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.

 

[Fredrik]

$g^{ab}$ 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.

 

[HallsofIvy]

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 $\frac{1}{g_{ij}}$.