I feel this must be an old question, but so far I have never seen it mentioned in any of the books I know. Most of modern treatment of advanced linear algebra is centered on tensor products and eigendecomposition, etc. So here is the question:
We know ordinary matrices provided a way of measuring a list of vectors $a_{i},i\in I_{n}$ such that for two different list of vectors we have the addition and multiplication. Further multiplication is associative, generates another list of vectors. Regardless of the specifics in row and column operations, I want to ask:
(1) Is it possible to define this for an array of vectors $a_{ij},i,j\in I_{n}$? I wish it to fulfill at least associativity, has an identity element, and makes some sense(such that respect to the individual $a_{ik}$ with $k$ fixed it resemble matrix multiplication?).
(2) Is it possible to define an exterior operation on it that resembling ordinary determinant?
(3) If the answer for above questions are both positive, is there any way to define a canonical decomposition of such $3$-dimensional matrices? I wish there be something resembling various decomposition theorems in ordinary linear algebra we have.
I had been thinking about this since I was a freshmen but I never got much progress (maybe I am too lazy). I will be happy to see if such an operation exists (or the reason why it should not exist).