Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $$A\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=3\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$$. Then the maximum value of $\operatorname{det}(\mathrm{A})$ is _________.

<p>Let $$A = \left[ {\matrix{ {{a_{11}}} & {{a_{12}}} & {{a_{13}}} \cr {{a_{21}}} & {{a_{22}}} & {{a_{23}}} \cr {{a_{31}}} & {{a_{32}}} & {{a_{33}}} \cr } } \right]$$</p> <p>Now</p> <p>$$A\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]=3\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]$$</p> <p>$$\left[ {\matrix{ {{a_{11}}} & {{a_{12}}} & {{a_{13}}} \cr {{a_{21}}} & {{a_{22}}} & {{a_{23}}} \cr {{a_{31}}} & {{a_{32}}} & {{a_{33}}} \cr } } \right]\left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right] = \left[ {\matrix{ 3 \cr 3 \cr 3 \cr } } \right]$$</p> <p>$$\begin{aligned} & a_{11}+a_{12}+a_{13}=3 \\ & a_{21}+a_{22}+a_{23}=3 \\ & a_{31}+a_{32}+a_{33}=3 \end{aligned}$$</p> <p>Now for maximum value of $$\operatorname{det}(A)=a_{i j}\left\{\begin{array}{ll}0 & i \neq j \\ 3 & i=j\end{array}\right\}$$</p> <p>$\therefore|A|=27$</p>