Another evidence that linear algebra concepts are terribly confusing when layed out without geometric background.

This had been criticised for decades. Competently and popularly presented criticism goes back to at least as far as 1957 (Artin's Geometric Algebra; see discussion of determinants somewhere near the beginning) but linear algebra is still often presented decoupled from geometry.

I wonder though if there's purely algebraic approach to matrices that explains as much (or more) as geometric one. Maybe approaching algebra of matrices consistently as an example of category algebra could be illuminating.