If it is also allowed that the software itself is poorly written ;-) then you might be interested in my matrix-calculator MatMate. Here we have much less functionality to describe than in Maxima et al. - just a simple, but nontrivial matrix "pocket" calculator.
Because there are many errors you have even the possibility to include criticism concerning technology/ implementation and concerning the syntax/ semantic-paradigms (which may be interesting for someone who is beginning a writer's carreer). http://go.helms-net.de/sw/matmate/index.htm The exisiting documentation is in german language but possibly gives an idea how to approach a not-too-big introduction/documentation.
(well, even if this all sounds a bit funny/sarcastic when I reread my own msg - that was not intended but -perhaps- unavoidable. Instead maybe such a tiny software is indeed a better start than a full-featured one)