What comes first.. multiplication or transpose?

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?

Re: What comes first.. multiplication or transpose?

Quote:

Originally Posted by

**kmjt** 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?

Transpose, what you have means -2 times the transpose of AB.

CB

Re: What comes first.. multiplication or transpose?

if (-2)[(AB)^T] = (-2AB)^T, then why are you worrying about this? the map A-->A^T is linear.

Re: What comes first.. multiplication or transpose?

I may be asking a bit confusing. I am wondering would I multiply the -2 into the outcome of AB before determing the transpose, or other way around?

Re: What comes first.. multiplication or transpose?

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).