Newmann-Penrose Spin coefficients for Schwarschild metric

I need to use the N-P formalism to apply in my work so I'm trying first to apply in a simple case to understand better. So in this article ( https://arxiv.org/abs/1809.02764 ) which I'm using, they present a null tetrad for the Schwarszchild metric in pg.14 (with accordance with the Chandrasekhar tetrad in "mathematical theory of black holes pg.134-135):
$${e_{(A)}}^\mu=\frac{1}{\sqrt2}\begin{bmatrix}
1/X & 1 & 0 & 0\\
1 & -X & 0 & 0 \\
0&0&1/r&\frac{i}{rsin\theta}\\
0&0&1/r&-\frac{i}{rsin\theta}
\end{bmatrix},\;\;A=1,2,3,4;\;\;\mu=t,r,\phi,\theta;\;\;X=1-\frac{2M}{r}$$
For calculating the spin coefficient (one of them for example): $$\mu=\gamma_{243}=\frac{1}{2}\big(C_{243}+C_{432}-C_{324}\big);\;\;\;{C^D}_{AB}=({e^D}_{\alpha,\beta}-{e^D}_{\beta,\alpha}){e_A}^\alpha {e_B}^\beta$$
$$C_{ABC}=\eta_{AD}{C^D}_{BA}.$$
Then I found the inverse needed above:
$${e^{(A)}}_\mu=\frac{\sqrt2}{2}\begin{bmatrix}
X & 1 & 0 & 0\\
1 & -1/X & 0 & 0 \\
0 & 0 & r & -irsin\theta\\
0 & 0 & r & irsin\theta
\end{bmatrix}.$$
Now when I go for the calculation, I don't get to the mentionated results: $$\mu=-\frac{X}{\sqrt2r}$$.

The books change index meaning throughout the texts and I am very confused. In Chandrasekhar pg.81 there is a tetrad and the inverse and there I can reproduce the results, I just don't know why in the other case I'm getting wrong.

Firstly, it is important to note that the Newmann-Penrose (N-P) formalism is a powerful mathematical tool for studying the properties of spacetime, particularly in relation to black holes. It was developed by Roger Penrose and Ezra T. Newmann in the 1960s and has been widely used in the field of general relativity.

In the context of the Schwarschild metric, the N-P formalism involves the use of a null tetrad, which is a set of four linearly independent vectors that are all null (meaning they have zero length) and orthogonal to each other. This tetrad is used to decompose the spacetime into two null directions and two spacelike directions, allowing for a more manageable analysis of the metric.

In the article you have referenced, the authors have presented a null tetrad for the Schwarschild metric, which is in accordance with the Chandrasekhar tetrad. This tetrad is given by the matrix ${e_{(A)}}^\mu$, where A=1,2,3,4 and $\mu=t,r,\phi,\theta$. It is important to note that the indices A and $\mu$ have different meanings - A corresponds to the tetrad basis vectors, while $\mu$ corresponds to the spacetime coordinates.

To calculate the spin coefficient $\gamma_{243}$, the authors have used the formula: $$\gamma_{243}=\frac{1}{2}\big(C_{243}+C_{432}-C_{324}\big)$$ where $C_{ABC}$ is the N-P curvature coefficient, given by the formula ${C^D}_{AB}=({e^D}_{\alpha,\beta}-{e^D}_{\beta,\alpha}){e_A}^\alpha {e_B}^\beta$. Here, ${e^{(A)}}_\mu$ is the inverse of the tetrad matrix ${e_{(A)}}^\mu$, and is given by the matrix in your question.

It is important to note that the authors have used the Minkowski metric $\eta_{AB}$ (with signature +2) to calculate the N-P curvature coefficient, as shown in the formula $C_{ABC}=\eta_{AD}{C^D}_{BA}$. This is different from the metric used in the Chandrasekhar tetrad, which is the Schwarschild metric (with signature -2). Hence, the results obtained using the