Mathematics - Determinant Question with Solution | TestHub

$∆=\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|=\begin{array}{c}\underbrace{\left(a_1{b}_2{c}_3+a_2{b}_3{c}_1+a_3{b}_1{c}_2\right)}\\ x\end{array}-\begin{array}{c}\underbrace{\left(a_3{b}_2{c}_1+a_2{b}_1{c}_3+a_1{b}_3{c}_2\right)}\\ y\end{array}$
Now if$x\le 3$and$y\ge -3$
The$∆$can be maximum 6 but it is not possible
because$x=3\Rightarrow$each term of$x=1$
And$y=-3\Rightarrow$each term of$y=-1$
$\Rightarrow$$\prod _{i=1}^3{a}_i{b}_i{c}_i=1$and$\prod _{i=1}^3{a}_i{b}_i{c}_i=-1$, which is a contradiction.
Now for value to be$5$one the terms must be zero but that will make two terms zero which means answer cannot be$5$. Next possibility is$4$which can be obtained as$\left|\begin{array}{ccc}1& 1& 1\\ -1& 1& 1\\ 1& -1& 1\end{array}\right|=1\left(1+1\right)-1\left(-1-1\right)+1\left(1-1\right)=4$