Is there theory of determinants, rank of matrices and systems of linear equations with octonionic coefficients? Does anybody could indicate references? I want to know mainly does there exist a something similar to Cronecker-Capelli theorem?

Could you tell us more about your motivations ? A linear equation over octonions boils down to a linear equation over the real numbers, so that's it
– Adrien HardyAug 12 '11 at 8:36

I would like to echo the first part of @Adrien's comment. I am not an expert, and perhaps an expert will be able to point you to the right place. But I think this question would be improved with more background and motivation. mathoverflow.net/howtoask
– Theo Johnson-FreydAug 12 '11 at 9:11

This seems to be a difficult problem. Even over the quaternions, not much is known about solving either polynomial equations or solving systems of linear equations. Some results can be found in the work of Opfer, see his website: math.uni-hamburg.de/home/opfer/veroeffentlichungen.html Various other authors have looked at this for quaternions and other generalizations of complex numbers (e.g. Clifford algebras). However, none of the results available seem to go very deep. Of course, in practice one can do as Adrien suggests and simply rewrite everything in terms of real numbers.
– Hendrik De BieAug 12 '11 at 9:52

2

There is a good notion of determiant of Hermitian octonionic matix of size 2 or 3, see e.g. math.ucr.edu/home/baez/octonions/oct.pdf . Using it, one can formulate a version of Sylvester criterion of positive definitness. Such determinant exists also for Hermitian quaternionic matrices of any size, it is called Moore determinant, see arxiv.com/PS_cache/math/pdf/0104/0104209v3.pdf and references there in.
– svaAug 12 '11 at 17:16

Have you tried to work out what happens with 1 or 2 unknowns and 1 or two equations? As mentioned above, you should be able to figure this out by brute force by converting everything back to quaternions, complex numbers, or, if necessary, reals.
– Deane YangAug 12 '11 at 18:47