What would I do first in a problem like this:
-2(AB)^T
Obviously first I work work AB. But then would I do the tranpose before multiple -2 in, or multiply -2 in before doing the transpose? I know they end up in the same answer but what way is correct? Which first?
if you get the same answer either way, then what is your concern?
suppose you multiply AB. now you have a matrix, let's call it M. if we multiply by -2 first, we have -2M. what is the ij-th entry of M? let's call it (gee, i am SO original) m_ij.
so when we multiply by -2, we now have -2(m_ij). now we take the transpose, and we have -2(m_ji) as our ij-th entry (it got multiplied by -2 as well, everything did).
ok, now let's do it "the other way". we have m_ij, and after we take the transpose, we have the ij-th entry is: m_ji. now we multiply by -2, and we get: -2(m_ji).
it looks to me like we get the same matrix either way.
i'll say it again, the transpose map is LINEAR. that means, in particular, that (cA)^T = c(A^T).