Let $x$ and $y$ be $3$vectors. For $i=1,2,3$, $y_i$ is the average
of $x_1, \ldots, x_i$. Then we have $y=Ax$, with

$A = \left[\begin{array}{ccc}1 & 1 & 1\\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right]$
Incorrect.

$A= \left[\begin{array}{ccc}1 & 0 & 0\\ 1/2 & 1/2 & 0 \\ 1/3 & 1/3 &1/3 \end{array}\right]$
Correct!

$A = \left[\begin{array}{ccc}1 &1/2 & 1/3\\ 0 & 1 & 1/2 \\ 0 & 0 & 1\end{array}\right]$
Incorrect.