I went to high school in Romania and first two years of college as well. We did tons of theory on about 2/3 (if not 3/4) of the material here (I also covered some of it when I continued my college in Montreal, Canada). But that took about 6 years of daily (maybe there was a day in the week when we didn't do math) study. And I'm sure we didn't cover it in the same depth as they do here (for most things we did a lot of proofs and theory). So it's possible but it takes a lot of time. In addition, it was usually motivated by the next step: you need to learn how to compute the inverse matrix in order to further process it with more complex algorithms which you actually use in labs or some sort of application (even if it's made up).

The interesting thing is that, for a Math student (or a college professor) this is the FOUNDATION of math you need in order to study advanced stuff. For some of the recent math proofs (think famous ones) you'd need to study a whole year (after you know most of this stuff) just to learn the theory foundations for that proof.

I'm of course exaggerating a bit - most Math majors would probably go through most of the stuff in this PDF but not necessarily all (machine learning stuff for example, and possibly numerical analysis stuff) of it. And if they do they usually forget* most of it, except the areas they use in their daily research (assuming they got for an advanced degree). And when you use some of this stuff every day it becomes second nature after a certain point.

My experience is mostly based on the Romanian educational system which is typically more hardcore science (Math and Physics) even for Computer Engineering degrees.

*Sometimes you don't forget stuff, you just don't think about it until you see it again (concrete example for me: Schur decomposition) and then partial memories of it come back.