Chapter 1, Problem 21SE

Prove: If A is an m × n matrix and B is the n × 1 matrix each of whose entries is 1/n, then

$AB=\left[\begin{array}{l}\overline{r_1}\\ \overline{r_2}\\ ⋮\\ \overline{r_m}\end{array}\right]$

where $\overline{r_i}$ is the average of the entries in the ith row of A.