You are not supposed to use eigenvectors to solve this question since my textbook hasn't even brought it up yet (the questions are for a chapter that is before eigenvectors are even introduced).

For example, how do you figure out what this transformation matrix does (standard basis):

$\displaystyle \vec e A = \vec e \frac{1}{3}\left(\begin{array}{ccc}-2&1&1\\1&-2&1\\1&1&-2\end{array}\right)$

I tried changing it to another basis, $\displaystyle \vec f = \vec e \left(\begin{array}{ccc}1&-2&1\\1&1&-2\\1&1&1\end{array}\right)$, and got this matrix instead: $\displaystyle B = \left(\begin{array}{ccc}0&0&0\\0&-1&0\\0&0&-1\end{array}\right)$