How to Find the General Element of (AB)^T for (AB)^T [AB transposed]

[qwerty5]

Homework Statement

Write the general element in terms of aij and bij for (AB)^T [AB transposed].

Homework Equations

(AB)^T = B^T*A^T; A=[aij]mxn; B=[bij]nxp

The Attempt at a Solution

n
AB= [sigma aik*bkj]mxp. Let this be equal to [xij]mxp
k=1
n
(AB)^T=[[sigma aik*bkj]mxp]^T
k=1

=[xji]pxm

n
=[sigma aki*bjk]mxp
k=1
n
so the general element xji=[sigma aki*bjk]
k=1


My teacher says this is wrong. Where did I go wrong?


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Alternate way I used to "check" my wrong answer:
n
(AB)^T=B^T*A^T=[bji]pxn[aji]nxm=[sigma bjk*aki]pxm=[sigma aki*bjk]pxm
k=1


We are using an differential equations/linear algebra textbook for engineers. It never discusses element-by-element proofs, and it leaves out many important differential equations topics, such as exact equations. I have a real diff eq book that my neighbor lent me, but I have to teach myself these types of problems through Wikipedia.

 

[Kreizhn]

I'm really not certain what all that stuff that's not in LaTeX is, but consider this:

If A = \{a_{ij} \} then you know that 1) A^T = \{ a_{ji} \}
and you also know that 2) (AB)^T = B^T A^T.

Now if B = \{ b_{jk} \} (where I've used the index "j" again since I know that j will iterate B in precisely the same manner as A for AB to make sense) you can write the matrix AB as AB = \displaystyle \left\{ a_{ij} b_{jk} \right\}. Now let's say, that without being too rigorous, you were to apply the operations from 1) and 2) to this sum, what would you get?

 

[qwerty5]

[bkj][aij] ?

 

[max111]

So I let A={a_{}ij} and B={b_{}ij}.

Then I know that A^{}T={a_{}ji} and B^{}T={a_{}ji}.

A typical element of the product B^{}TA^{}T={b_{}jia_{}ji}.

However, B^{}TA^{}T={\Sigma^{}n_{}k=1b_{}jka_{}kj}.


Is this correct?

 

[max111]

So I let A={a_{}ij} and B={b_{}ij}.

Then I know that A^{}T={a_{}ji} and B^{}T={a_{}ji}.

A typical element of the product B^{}TA^{}T={b_{}jia_{}ji}.

The sum would therefore be: B^{}TA^{}T={\Sigmab_{}jka_{}ki} from k=1 to n.

Is this correct?

 

[qwerty5]

Max, I think we are both right.

What did the professor get?