I have been learning how to transform from one basis to another. I don't have any issues when both basis are orthogonal because I use a different formula then the one below. An issue I am having is when the new basis is not orthogonal. From searching the internet, I have the following formula for finding an inverse of a matrix (Assume 3 dimensions):
$$ det(\textbf{A}) = \epsilon_{ijk}A_{i1}A_{j2}A_{k3}$$
$$A^{-1}_{ij} = \frac{1}{2*det(\textbf{A})} \epsilon_{jmn}\epsilon_{ipq}A_{mp}A_{nq}$$
Here is where I am missing something. When I calculate the determinant, I get a scalar (number), which is what I expect. Lets say the determinant was 4. The issue is when I try to calculate the inverse. I get: $$A^{-1}_{ij} = \frac{1}{2*4} \epsilon_{jmn}\epsilon_{ipq}A_{mp}A_{nq}$$
The $\epsilon_{jmn}\epsilon_{ipq}A_{mp}A_{nq}$ will just produce another number without basis vectors attached to it. Does anyone know where I am making a mistake, or did the formula leave out the basis vectors? If they did, can you please let me know which indicies they should have? I expect its an outer product of two, but I don't know which.
Thank you!