Let's concentrate in the 2 dimensional case. I'm looking for a $2 \times 2$ Matrix $A$ that for a vector $x=(x_1, x_2)$ will satisfy: $$ xAx^t = \frac{x_1+x_2}{2}$$
Using simple methods of quadratic forms, I was able to find a matrix that returns the $\text{(average)}^2$ of a vector. which is: $$ B = \begin{bmatrix} 0.25 & 0.25 \\ 0.25 & 0.25 \\ \end{bmatrix}$$ And indeed if $x=(5,9)$ than $ xBx^t = 49 = 7^2$. And if I not mistaken, the general case $n$ dimensions is as simple to solve. we will end up with a matrix where $a_{ij} = \frac{1}{n^2}$ for every $0 \leq i,j \leq n$
Now, when I try to create a matrix that returns the mean, I encounter some problems. I know that if such a matrix indeed exist, I think it should be in the form: $$ A = \begin{bmatrix} a & b \\ b & a \\ \end{bmatrix}$$ and we can also understand that $2a+2b$ = $1$ than it appears we can try and solve the equation for $a$ to see if such an $a$ exist. But I just have to think there has to be simple way to find the Matrix if it exsist, or to tell pretty fast that there is no such a matrix.
No such matrix exists, because the scaling is wrong. If you replace the vector $x$ by the vector $\lambda x$, the right side is multiplied by $\lambda$; however, the left side is multiplied by $\lambda^2$.