Proving Columns of (AB) Sum to st

[oddiseas]

Homework Statement

Let A be an m*p matrix whose columns all add to the same total say "s". Let B be a p*n matrix whose columns alladd to the same total say "t".Prove that the n columns of (AB) all sum to st.

Homework Equations

This is an obvious result but i am having trouble proving it with summation notation which is what we were told to do.

A=∑(i=1 to m)a_{ij}

B=∑(j=1 to p)b_{jk}

(AB)=∑(j=1 to p)a_{ij}*b_{jk}

But i am having trouble proving that each column sums to st even though i know this is the case.

 

[lippyka]

The Attempt at a SolutionLet the jth column of A be denoted by [a_1j, a_2j ... a_mj]. Let the kth column of B be denoted by [b_1k, b_2k ... b_pk]. Then the kth column of AB is given by [a_11*b_1k + a_12*b_2k + ... + a_m1*b_pk, a_12*b_1k + a_22*b_2k + ... + a_m2*b_pk, ... , a_1m*b_1k + a_2m*b_2k + ... + a_mm*b_pk]To prove that each column sums to st, we have to show that a_11*b_1k + a_12*b_2k + ... + a_m1*b_pk + a_12*b_1k + a_22*b_2k + ... + a_m2*b_pk + ...+ a_1m*b_1k + a_2m*b_2k + ... + a_mm*b_pk = stThis is where i am stuck. I can see that the proof is obvious but do not know how to express it with summation notation. Any help would be greatly appreciated!