## anonymous 5 years ago Prove that whenever A and B are matrices for which AB is defined, then (B^T)(A^T) is also defined. Then show (AB)^T=(B^T)(A^T). (transpose)

1. anonymous

So what does it mean AB is defined?

2. anonymous

I'm guessing, if A is an m x n matrix and B is an n x p matrix

3. anonymous

...then AB will be defined as the m x p matrix

4. anonymous

Well my book says the transpose of a product of any number of matrices is the product of the tranposes in the reverse order.

5. anonymous

my prof always starts of a proof asking "What do we know"

6. anonymous

So we know that AB is defined

7. anonymous

I guess i'm having trouble understanding this whole "ijth" entry thing, I didn't understand it much in lecture and my book doesn't expand

8. anonymous

can you tell me what topic it is?

9. anonymous

n-Dimensional Geometry: Matrix Multiplication the book says that the matrix product AB is the mxp matrix whose ijth entry is the dot product of the ith row of A and the jth column of B (considered as vectors in R^n)

10. anonymous

check out this webpage it explains matrix multiplication in easy to understand language http://www.purplemath.com/modules/mtrxmult.htm

11. anonymous

thank you!

12. anonymous

you're welcome

13. anonymous

did that help?

14. anonymous

a little, i looked into my book a bit more and convinced myself that the ijth entry of AB is $\sum_{k}^{n}$ $a _{ik}b _{kj}$ , which equals the jith entry of $B ^{T}A ^{T}$ which is $\sum_{k}^{n} b _{kj}a _{ik}$