[meuk]

It recently occurred to me that if you use that matrices represent linear functions, you don't have to do tedious math to prove that matrix multiplication is associative (that is, (A * B) * C = A * (B * C), which allows us to write A * B * C without brackets, since it doesn't matter how we place the brackets anyway).

For a matrix M, denote f_M(x) = M * x. Then f_{A * B} = f_A(f_B(x)) so that f_{(A * B) * C} = f_{A * B}(f_C(x)) = f_A(f_B(f_C(x))) and also f_{A * (B * C)} = f_A(f_{B * C}(x)) = f_A(f_B(f_C(x))).

So f_{(A * B) * C} = f_{A * (B * C)} = f_A(f_B(f_C(x)))