# Show that matrix AB has its row and column vectors in A and B

1. Apr 26, 2009

### jheld

1. The problem statement, all variables and given/known data
Given matrices A and B, show that the row vectors of AB are in the row space of B an the column vectors of AB are in the column space of A

2. Relevant equations
Just matrix multiplication, reduced row echelon form, and leading one's for row and columns

3. The attempt at a solution
I am unsure if I need to find the two separate matrices, or if this is just a general problem.
I understand what they are asking, but I am unsure of how to find the right matrices, and what it means to be "in" the row and column vectors. I think it means that they are in the correct rows and columns from the respective single matrices A and B.

2. Apr 26, 2009

### Dick

It means that a row of AB is a linear combination of rows of B and a column of AB is a linear combination of columns of A. All you need is the definition of matrix multiplication, A_{ij}*B_{jk}=(AB)_{ik}.