Let A be a 3 × 3 matrix of non-negative real elements such that A [ 1 1 1 ] = 3 [ 1 1 1 ] . Then the maximum value of d e t ( A ) is _______. [2024]

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Solution

Q.
Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $A [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = 3 [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] .$ Then the maximum value of $d e t (A)$ is _______. [2024]

Ans.
(27)

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

Now, $A [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = 3 [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] \Rightarrow [\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}] [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = [\begin{matrix} 3 \\ 3 \\ 3 \end{matrix}]$

$a_{11} + a_{12} + a_{13} = 3$

$a_{21} + a_{22} + a_{23} = 3$

$a_{31} + a_{32} + a_{33} = 3$

Now, for maximum value of $d e t (A) = a_{i j} = {\begin{matrix} 0, & i \neq j \\ 3, & i = j \end{matrix}$

$∴$ $| A | = 27$

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