The article by J.-L. Dorier in <A HREF="http://books.google.nl/books?id=afW8TGIMuigC">On the Teaching of Linear Algebra</A> suggests the answer to your question will be different for the UK and for continental Europe:

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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 <A HREF="https://archive.org/details/corsodianalisia00garbgoog">Corso di Analisi Algebrica</A> from **1886**. This is part 1 of the course, called "Introductory theories", so it may well have been intended for undergraduates.

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