First a little history
The procedure for solving simultaneous linear equations now called Gaussian elimination appears in the ancient Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen problems, with two to five equations.
This is discussed in Roger Hart, The Chinese Roots of Linear Algebra; however, in Euope
Systems of linear equations arose with the introduction in 1637 by René Descartes of coordinates in geometry. In fact, in this new geometry, now called Cartesian geometry, lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations. However, the first systematic methods for solving linear systems used determinants, first considered by Leibniz in 1693.
In fact, Leibniz considered there to be a theory of 'extension' or 'logical characteristic' but wasn't able to come up with a viable such theory; in 1844 a prize competition was instituted on exactly this problem; this was won by Grassmann who had entered an essay 'Geometrische analsye...' after being persuaded by Mobius to enter; this included foundational new topics of what is today called linear algebra.
It was around this time (actually 1843), that Hamilton discovered the quaternions which prompted the discovery of other hyper-complex systems and then five years later, the English mathematician, James Joseph Sylvester introduced the term matrix (which is Latin for womb); it was another English mathematician, William Clifford who combined both Grassmanns theory and the theory of hyper-complex systems into what are now known as Clifford algebras.
In the transition from the early quantum mechanics to conventional quantum mechanics, Heisenberg and Jordan rediscovered matrix multiplication in 1925 (although Connes says that this would be better understood through groupoids).
It was Emmy Noether and her school who pioneered the study of abstract algebraic structures per se placing them in a systematic foundation; and in 1930, Van der Waerden published his Modern Algebra which 'forever changed' how algebra was taught in universities.
I would consider that it was all of these developments that pushed the undergraduate curriculum to the consideration of abstract algebra per se and of intrinsic structures and not just quantum mechanics.
(On a personal note, matrix mathematics wasn't just taught at universities, I distinctly remember being taught matrix mathematics at school).