Let A be a m * p matrix whose columns all add to the same total s; and B be a p * n matrix whose columns all add to the same total t: Using summation notation prove that the n columns of AB each total st:
By definition of matrix multiplication, the entries of $\displaystyle AB$ are given by $\displaystyle \sum_{i=1}^p a_{j,i} b_{i,k}$.
We seek the sum of an arbitrary column of $\displaystyle AB$, given by $\displaystyle \sum_{j=1}^m\left(\sum_{i=1}^p a_{j,i} b_{i,k}\right)$
$\displaystyle =\sum_{i=1}^p \left( b_{i,k} \sum_{j=1}^m a_{j,i}\right)$
$\displaystyle =\sum_{i=1}^p \left( b_{i,k} s\right)$
$\displaystyle =s\sum_{i=1}^p b_{i,k} $
$\displaystyle =st $.