yes, you can "FOIL" matrices:

(A+B)(C+D) = AC + BD + AD + BC, but remember the order of multiplication in each term is important:

(A+B)(A-B) = A^{2}+ BA - AB - B^{2}, NOT A^{2}- B^{2}(because usually AB ≠ BA).

matrix differentiation is just a convenient way of keeping track of the various partial derivatives involved. you might want to look here:

Matrix calculus - Wikipedia, the free encyclopedia