​question 8
Q.8. 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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Please find below the solution to the asked query:

$$\begin{gathered} △=\left|\begin{array}{ccc}{\mathrm{a}}_1& {\mathrm{a}}_2& {\mathrm{a}}_3\\ {\mathrm{b}}_1& {\mathrm{b}}_2& {\mathrm{b}}_3\\ {\mathrm{c}}_1& {\mathrm{c}}_2& {\mathrm{c}}_3\end{array}\right| \\ \mathrm{On} \mathrm{expansion} \mathrm{we} \mathrm{get} \\ \Rightarrow △={\mathrm{a}}_1\left({\mathrm{b}}_2{\mathrm{c}}_3-{\mathrm{b}}_3{\mathrm{c}}_2\right)-{\mathrm{a}}_2\left({\mathrm{b}}_1{\mathrm{c}}_3-{\mathrm{b}}_3{\mathrm{c}}_1\right)+{\mathrm{a}}_3\left({\mathrm{b}}_1{\mathrm{c}}_2-{\mathrm{b}}_2{\mathrm{c}}_1\right) \\ ={\mathrm{a}}_1{\mathrm{b}}_2{\mathrm{c}}_3-{\mathrm{a}}_1{\mathrm{b}}_3{\mathrm{c}}_2-{\mathrm{a}}_2{\mathrm{b}}_1{\mathrm{c}}_3+{\mathrm{a}}_2{\mathrm{b}}_3{\mathrm{c}}_1+{\mathrm{a}}_3{\mathrm{b}}_1{\mathrm{c}}_2-{\mathrm{a}}_3{\mathrm{b}}_2{\mathrm{c}}_1 \\ \mathrm{As} \mathrm{there} \mathrm{are} 6 \mathrm{terms} \mathrm{and} \mathrm{each} \mathrm{element} \mathrm{of} \mathrm{matrix} \mathrm{is} \mathrm{from} \mathrm{set} \left\{-1,0,1\right\} \\ \mathrm{hence} \mathrm{maximum} \mathrm{value} \mathrm{can} \mathrm{be} 6 , \mathrm{when} \mathrm{all} \mathrm{six} \mathrm{terms} \mathrm{are} 1 \\ \mathrm{Value} \mathrm{can} \mathrm{be} 6 \mathrm{when} \\ {\mathrm{a}}_1=1, {\mathrm{a}}_2=-1, {\mathrm{a}}_3=1, {\mathrm{b}}_2{\mathrm{c}}_3={\mathrm{b}}_1{\mathrm{c}}_3={\mathrm{b}}_1{\mathrm{c}}_2=1 \\ {\mathrm{b}}_3{\mathrm{c}}_2={\mathrm{b}}_3{\mathrm{c}}_1={\mathrm{b}}_2{\mathrm{c}}_1=-1 \\ \Rightarrow \left({\mathrm{b}}_2{\mathrm{c}}_3\right)\left({\mathrm{b}}_3{\mathrm{c}}_1\right)\left({\mathrm{b}}_1{\mathrm{c}}_2\right)=-1 \mathrm{and} \left({\mathrm{b}}_1{\mathrm{c}}_3\right)\left({\mathrm{b}}_3{\mathrm{c}}_2\right)\left({\mathrm{b}}_2{\mathrm{c}}_1\right)=-1 \\ \mathrm{which} \mathrm{gives} \\ {\mathrm{b}}_1{\mathrm{b}}_2{\mathrm{b}}_3{\mathrm{c}}_1{\mathrm{c}}_2{\mathrm{c}}_3=-1 \mathrm{and} 1 \\ \mathrm{which} \mathrm{is} \mathrm{not} \mathrm{possible}. \end{gathered}$$



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