I would think the source to consult is On the Teaching of Linear Algebra by J.-L. Dorier. I can only read parts of it on Google Books, but here is one quote that suggests the answer to your question will be different for the UK and for continental Europe:
http://ilorentz.org/beenakker/MO/linearalgebra.png
In an attempt to answer your question more directly, I have searched for early University text books that introduce matrix multiplication. It was introduced in the context of the theory of determinants, to write the product of two determinants as a single determinant, in the text Corso di Analisi Algebrica from 1886. This is part 1 of the course, so it may well be intended for undergraduates.