[Garlef]

It's been a few years (~ 15) since I read it.

The determinant trick I'm referring to is used in prop 2.4 (in the Version I found via Google).

They use the determinant of the adjugate matrix if I remember correctly... But they just hit the reader with this without any motivation or even naming it.

[wbl]

For Nakayama lemma (but they give a cleaner proof later). Note that the book was from the 1960's and from lectures at Oxford: matrix manipulations like that were a lot more commonplace.