Collapsed Adjacency Matrices and Character Tables of rank 2

If G is acting 2-transitive on a finite set M of order n > 1, the subdegrees of G are 1 and n − 1. The collapsed adjacency matrices are the identity matrix of degree 2 and the following matrix:

( \begin{matrix}0 & n − 1 \\ 1 & n − 2 \end{matrix} ) .  (1)

The eigenvectors belonging to this matrix are

[\frac{1}{n − 1}, 1], [−1, 1]  (2)

Finally the character table is the following:

\begin{aligned}& S_{1} & S_{2} \\ χ_{1} & 1 & n − 1 \\ χ_{2} & 1 & −1 \end{aligned}  (3)