One reason linear algebra is so useful is that the basic notions, like rank, have so many equivalent definitions. Some are better for formulating problems, some for proving theorems, and some for doing computations. The ability to freely move between these is the key to solving many problems.
Some of these definitions of course won't make sense for hypermatrices, but many of them do. The problem is that they don't usually end up being equivalent.
For example: you can define rank one hypermatrices as outer products of vectors ("simple tensors") and define the minimum number of such terms which must be summed to yield a given hypermatrix to be the "hyperrank". But this does not properly classify all hypermatrices up to changes of basis along all the "axes" of the hypermatrix as it does for matrices (in fact the number of equivalence classes is no longer even finite). And except in very simple cases this does not agree with what you'd get by the vanishing of certain "hyperdeterminants" -- indeed, the set of hypermatrices with hyperrank at most $k$ hypermatrices isn't even closed.
So the things you can compute aren't the same as the things you'd like to compute, and everything ends up feeling much more ad hoc.
Of course this complexity makes for a lot of interesting things to study, but not a simple widely applicable tool every undergrad should learn. Though perhaps they should learn why they don't learn it!