Let me tell what I know about this. It is well-known that Heisenberg invented matrix multiplication himself, in his great paper that is considered part of the foundation of quantum mechanics. This was in 1925, and the story is very well documented. Then very shortly after that Born and Jordan recognized that this is matrix multiplication, BECAUSE one of them had a course on "hypercomplex numbers" as a student.
The clear conclusion that I make of this is that in the first decade (when they were all students) of 20th century matrix multiplication was not taught to students on a regular basis in the best European universities.
On the other hand at the present time matrices are taught to ALL (science) undergraduates; this is even more standard than Calculus (I judge from my experience in Soviet Union and USA, but I suppose this is the case everywhere).
So my questions are:
- When did this dramatic transition in undergraduate curriculum happen?
and even more interestingly:
- Why did it happen?
On the second question I have a conjecture: it is exactly because of the invention of quantum mechanics. I have some supporting evidence and "philosophical arguments" in favor of this. But to investigate this matter, it is good first of all to find the answer on the first question.
I know that matrix multiplication was probably introduced by Cayley, but it is a very long way for a new mathematical object to the undergraduate curriculum, and most of our inventions never make it this way :-)
A similar question is posted on MO.