I'm familiar with sympy, the matlab symbolic package, reduce, and have tried out a few other computer algebra systems. However, as far as I can tell, none of them seem to be able to do algebra on variable sized matrices - they can only work with fixed sized matrices.

Are there any that can do algebra for variable sized matrices? I understand there would be quite a few gross cases but I feel like there is a significant amount that is doable simply because of the ease of many simplifications/algebra one can do by hand in with matrices in R^nxn.

It is possible to just work with non-communiative algebraic elements in many of these, and so that covers addition and Hadamard product with matrices, which is useful and a start. However it covers a very small portion of what one actually does with matrices (say, transpose, inverses, eigenvalue decomposition, using matrices in R^nxm, etc.). Does any more general software exist?

(this question is reposted from here because mathoverflow seemed like a more appropriate place to ask this question)