# Is it possible to do matrix algebra symbolically?

I'm reading a book that has the following theorem:

I am trying to make Mathematica perform these calculations for me. Is it possible to do it without having to declare the matrices? For example, I tried to write: Transpose[A.B]. I expected it to return $B^TA^T$ or something similar, but it actually doesn't do that. At least in a trivial way.

• It may be possible to carry out symbolic matrix algebra once you provide appropriate assumptions: see e.g. Matrices and, more generally, Symbolic Tensors. – MarcoB Dec 7 '16 at 6:35
• – MarcoB Dec 7 '16 at 6:45
• Take a look at NCAlgebra and if you can do the above calculations with it, please post a self-answer. – Szabolcs Dec 7 '16 at 9:38