A million times this. Its not trivial to geometrically understand linear algebra without getting into topology if you go "determinants first". An algebraic understanding of linear algebra is not the most adequate if you are learning it to further understand topics such as machine learning or signal processing.

PD: If you really do like or need the algebraic approach, Shilov's book is the way to go after this course... I think the first chapter is about determinants and he builds the rest of the book from there. It also includes a chapter on tensor algebra.