In the text, just before the 2D example, it says
“The relation \textbf{a}_i \cdot \textbf{b}_j=2πδ_{ij} means that if we write the lattice vectors as rows of a matrix, the reciprocal lattice vectors are 2\pi times the rows of the inverse of that matrix.”
I’m quite sure this should be that the reciprocal lattice vectors are 2\pi times the \textbf{columns} of the inverse of that matrix.