I am trying to verify that the inverse KK metric listed on the KK Wikipedia page is in fact the inverse of the KK metric. To do this, I am multiplying them with matrix multiplication to see if the identity matrix results. However, there is some discrepancy that I am unable to reconcile. Also, I am 5D and my index counting convention goes from 1 to 5, with no 0.
The KK metric and inverse metric are:
$$ \Sigma_{AB}=\left(\begin{matrix} -1-B_1B_1\chi^5 & -B_1B_2\chi^5 & -B_1B_3\chi^5 & -B_1B_4\chi^5 & B_1\chi^5\\[8pt] -B_2B_1\chi^5 & 1-B_2B_2\chi^5 & -B_2B_3\chi^5 & -B_2B_4\chi^5 & B_2\chi^5\\[8pt] -B_3B_1\chi^5 & -B_3B_2\chi^5 & 1-B_3B_3\chi^5 & -B_3B_4\chi^5 & B_3\chi^5\\[8pt] -B_4B_1\chi^5 & -B_4B_2\chi^5 & -B_4B_3\chi^5 & 1-B_4B_4\chi^5 & B_4\chi^5\\[8pt] B_1\chi^5 & B_2\chi^5 & B_3\chi^5 & B_4\chi^5 & \chi^5\\[8pt] \end{matrix}\right) $$
and
$$ \Sigma^{AB}=\left(\begin{matrix} -1 & 0 & 0 &0 & -B^1 \\[8pt] 0 & 1 & 0 &0 & -B^2 \\[8pt] 0 & 0 & 1 &0 & -B^3 \\[8pt] 0 & 0 & 0 &1 & -B^4 \\[8pt] -B^1 &-B^2 &-B^3 &-B^4 &\eta_{\sigma\rho}B^\sigma B^\rho +\dfrac{1}{\chi^5} \end{matrix}\right) $$
Then when I begin to multiply to get the first row, first column value of the product matrix, which should be 1 if it is to be the identity matrix, I get a sign problem.
Following the rules of matrix multiplication I get
$$ 1+B_1B_1\chi^5+B_1\chi^5(-B^1) $$
If I say that $B_1=B^1$ it will work, but actually using the regular formula
$$ A^\mu=\eta^{\mu\nu}A_\nu $$
Then due to the single minus in the Minkowski signature, I should have $B^1=-B_1$ but then the one matrix is not the inverse of the other.
Is this my error or did the source of the inverse metric I found use sloppy labeling with the upper and lower indices?