$A=\left[\begin{array}{lll}\frac{1}{6}& \frac{-1}{3}& \frac{-1}{6}\\ \frac{-1}{3}& \frac{2}{3}& \frac{1}{3}\\ \frac{-1}{6}& \frac{1}{3}& \frac{1}{6}\end{array}\right]$

$\Rightarrow A^2=A\times A=\left[\begin{array}{ccc}\frac{1}{6}& \frac{-1}{3}& \frac{-1}{6}\\ \frac{-1}{3}& \frac{2}{3}& \frac{1}{3}\\ \frac{-1}{6}& \frac{1}{3}& \frac{1}{6}\end{array}\right]\times \left[\begin{array}{ccc}\frac{1}{6}& \frac{-1}{3}& \frac{-1}{6}\\ \frac{-1}{3}& \frac{2}{3}& \frac{1}{3}\\ \frac{-1}{6}& \frac{1}{3}& \frac{1}{6}\end{array}\right]$

$=\frac{1}{36}\left[\begin{array}{ccc}6& -12& -6\\ -12& 24& 12\\ -6& 12& 6\end{array}\right]$

$=\frac{6}{36}\left[\begin{array}{ccc}1& -2& -1\\ -2& 4& 2\\ -1& 2& 1\end{array}\right]\begin{array}{}\\ \end{array}$

$=\left[\begin{array}{ccc}\frac{1}{6}& \frac{-2}{6}& \frac{-1}{6}\\ \frac{-2}{6}& \frac{4}{6}& \frac{2}{6}\\ \frac{-1}{6}& \frac{2}{6}& \frac{1}{6}\end{array}\right]$

$A^2=A$

$\therefore A^{2016l}=A, A^{2017m}=A, A^{2018n}=A$

$\mathrm{LHS}: A+A+A=3A$

$\mathrm{RHS}: \frac{1}{\alpha }A$

$\therefore \alpha =\frac{1}{3}$