Let P be a matrix of order 3 times 3 such tha | vedclass

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(D) Let $P = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix}$ where $a_{ij} \in \{-1, 0

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$4$

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(D) Let $P = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix}$ where $a_{ij} \in \{-1, 0, 1\}$.The determinant of a $3 	imes 3$ matrix is given by the sum of $6$ products of $3$ elements each.For a $3 	imes 3$ matrix with entries in $\{-1, 0, 1\}$,the maximum value of the determinant is known to be $4$. This is a specific result related to Hadamard's maximum determinant problem for matrices with entries in $\{-1, 1\}$,but with $0$ allowed,the maximum remains $4$ for $n=3$.To see why $4$ is the maximum,consider the matrix:$P = \begin{bmatrix} 1 & -1 & 0 \ 1 & 1 & 1 \ -1 & -1 & 1 \end{bmatrix}$Calculating the determinant:$\det(P) = 1(1 - (-1)) - (-1)(1 - (-1)) + 0(-1 - (-1))$$\det(P) = 1(2) + 1(2) + 0 = 4$.Thus,the maximum possible value is $4$.

Let $P$ be a matrix of order $3 \times 3$ such that all the entries in $P$ are from the set $\{-1, 0, 1\}$. Then,the maximum possible value of the determinant of $P$ is:

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