Thread: Linear algebra proof

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:

Originally Posted by ulysses123

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 $.

Thanks, what i was doing was expressing one column in Matrix A, by the sum of all columns divided by p, which was making it difficult to then simplify the expression for AB.