TL;DR: In the US, matrices and matrix multiplication probably became an expected part of the curriculum for engineering majors ca. 1970, and I would guess that the reason for this was its applications in electrical engineering and materials science.
19th Century: Through the 19th century, most physicists customarily wrote all equations component-wise. This covers Maxwell, and seems to have continued at least as late as 1905, because Einstein wrote his original paper on relativity in very awkward notation, without the use of vectors. There was a period when some theorists in physics were pushing a quaternionic system, but it was never widely used by physicists, much less by working electrical engineers. Hamilton's huge book on quaternions was known as the book that lots of people owned but nobody ever read.
Early 20th Century: The influential vector presentations of physics began when Gibbs and Heaviside simplified the useful aspects of the quaternionic system into the modern vector-scalar system, which was published in an influential 1901 text by Wilson. However, this seems to have taken a very long time to filter down into the curriculum. For instance, I have a copy of the 1927 edition of Elements of Physics, by Millikan and Gale. I believe this was the most popular introductory physics text in the US at that time. The word "vector" never appears, and we have on p. 64, "A force is completely described when its magnitude, its direction, and the point at which it is applied are given." In other words, the definition is such that forces don't even obey the ordinary law of vector addition, because there is a third datum, the point of application. There is no conception being presented of a more general abstract algebraic concept, just a specific definition of a force.
Middle 20th Century: I checked out some old UC Berkeley catalogs in order to see what the later historical situation was in the US.
In the 1959-60 catalog, there are two linear algebra courses in the math department:
- a lower-division course of linear algebra for social science majors
- an upper-division course that is part of a one-year algebra course for math majors
There were no linear algebra courses oriented at physical scientists or engineers. The engineering programs also did not offer a linear algebra class; they required mathematics courses only as prerequisites to the required engineering classes, and those prerequisites included vector calculus and differential equations, but no linear algebra.
In the 1969-70 catalog, there is a lower division linear algebra course, Math 11C. However, none of the required courses for an engineering degree seem to have 11C as a direct or indirect prerequisite. Probably by this time many engineering students were taking 11C -- otherwise I don't see why it would need to be moved from upper division to lower division -- but it wasn't required.
It's implausible that this change ca. 1970 had anything to do with Heisenberg's discovery ca. 1927 that quantum mechanics could be done using matrices. Not only is the gap in time very long, but if you look at modern treatments of quantum mechanics in physics books aimed at sophomore engineering majors, they don't make use of any linear algebra. In fact, it's more common to see matrices popping up when mutual inductance and mutual capacitance are introduced, but I think most texts do this without using the word "matrix" and without ever introducing matrix multiplication.
Current situation in the US: Here is my impression of the current situation in the US, which differs from the comments in the middle of question 1. At my school, the math sequence taken by STEM students goes like this: calculus of functions of a single variable; functions of several variables, including div, grad, and curl and differential equations; linear algebra. The linear algebra course is placed at the very end, presumably on the theory that it's a higher priority for engineering students to learn vector calculus in time for their course in electromagnetism.
Students in the US may get exposed to matrices in a high school pre-calculus course, but I don't think this is standard or highly emphasized. I believe that math textbooks used for functions of several variables usually assume no knowledge of linear algebra, even though this often makes the exposition of the subject extremely awkward.