How would I simplify this matrix while calculating the determinant?
\begin{bmatrix} 2/3 & -1/6 & -1/3 \\ -1/3 & 5/6 & -1/3 \\ -1/3 & -1/6 & 2/3 \end{bmatrix}
The multiple is $1/18$ so
$$\frac{1}{18} \begin{bmatrix} 12 & -3 & -6 \\ -6 & 15 & -6 \\ -6 & -3 & 12 \end{bmatrix} $$
Using row reduction operations $-R_2 + R_3 \to R_3$ and $2R_2 + R_1 \to R_1$
$$\frac{1}{18} \begin{bmatrix} 0 & -27 & -18 \\ -6 & 15 & -6 \\ 0 & -18 & 18 \end{bmatrix} = \frac{6}{18}\begin{bmatrix} 27 & -18 \\ -18 & 18 \end{bmatrix} = \frac{6}{18}(27 \cdot 18 - 18 \cdot 18) = 54$$
Answer is $1/6$. Where did I go wrong? I still wish to complete this with whole numbers
$$\begin{bmatrix} 2/3 & -1/6 & -1/3 \\ -1/3 & 5/6 & -1/3 \\ -1/3 & -1/6 & 2/3 \end{bmatrix}=\frac{1}{18^3} \begin{bmatrix} 12 & -3 & -6 \\ -6 & 15 & -6 \\ -6 & -3 & 12 \end{bmatrix}=\frac{972}{18^3}=\frac{1}{6}$$