Question

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)

Answer 1

So what does it mean AB is defined?

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

So we know that AB is defined

Answer 7

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

Answer 8

can you tell me what topic it is?

Answer 9

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)

Answer 10

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

Answer 11

thank you!

Answer 12

you're welcome

Answer 13

did that help?

Answer 14

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 \[\sum_{k}^{n} b _{kj}a _{ik}\]