Let P be a matrix of order 3 × 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_____

Engineering

Mathematics

Introduction to Matrix

Question

Let P be a matrix of order 3 × 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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$| \begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} | = a_{1} (b_{2} c_{3} - b_{3} c_{2}) - a_{2} (b_{1} c_{3} - b_{3} c_{1}) + a_{3} (b_{1} c_{2} - b_{2} c_{1})$

$= a_{1} b_{2} c_{3} + a_{2} b_{3} c_{1} + a_{3} b_{1} c_{2} - a_{1} b_{3} c_{2} - a_{2} b_{1} c_{3} - a_{3} b_{2} c_{1}$

$\Rightarrow a_{1} = 1, a_{2} = 1, a_{3} = 1; b_{1} = - 1, b_{2} = 1, b_{3} = - 1; c_{1} = - 1, c_{2} = - 1, c_{3} = 1$

$| \begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix} | = 1 + 1 + 1 - 1 - (- 1) - (- 1) = 4 .$