Let $$A = \left[ {\matrix{ {{a_{11}}} & {{a_{12}}} & {{a_{13}}} \cr {{a_{21}}} & {{a_{22}}} & {{a_{23}}} \cr {{a_{31}}} & {{a_{32}}} & {{a_{33}}} \cr } } \right]$$

Now

$$A\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]=3\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]$$

$$\left[ {\matrix{ {{a_{11}}} & {{a_{12}}} & {{a_{13}}} \cr {{a_{21}}} & {{a_{22}}} & {{a_{23}}} \cr {{a_{31}}} & {{a_{32}}} & {{a_{33}}} \cr } } \right]\left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right] = \left[ {\matrix{ 3 \cr 3 \cr 3 \cr } } \right]$$

$$\begin{aligned} & a_{11}+a_{12}+a_{13}=3 \\ & a_{21}+a_{22}+a_{23}=3 \\ & a_{31}+a_{32}+a_{33}=3 \end{aligned}$$

Now for maximum value of $$\operatorname{det}(A)=a_{i j}\left\{\begin{array}{ll}0 & i \neq j \\ 3 & i=j\end{array}\right\}$$

$$\therefore|A|=27$$