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I'm just putting down some thoughts here that hopefully someone else will know about.

sage: M = matrix([[2,3],[3,2]])
sage: e^M
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
TypeError: mutable matrices are unhashable
sage: exp(M)
[1/2*(e^6 + 1)*e^(-1) 1/2*(e^6 - 1)*e^(-1)]
[1/2*(e^6 - 1)*e^(-1) 1/2*(e^6 + 1)*e^(-1)]
sage: M.exp()
[1/2*(e^6 + 1)*e^(-1) 1/2*(e^6 - 1)*e^(-1)]
[1/2*(e^6 - 1)*e^(-1) 1/2*(e^6 + 1)*e^(-1)]

Naturally, this is really because there isn't any sin method. However:

sage: M = matrix([[2,3],[3,2]])
sage: M.simp[tab] # nothing
sage: M = matrix(SR,[[2,3],[3,2]])
sage: M.simp[tab]
M.simplify           M.simplify_rational  M.simplify_trig

There still isn't a sin. But this could be doable. Of course, the question remains as to how many such things we would want to add. Thoughts?