Question

If the value of the determinant $\left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|$ is positive, then

• A. $abc<-2$
• B. $abc>0$
• C. $abc>-8$
• D. $abc=0$

Answer 1

The correct option is C $abc>-8$

$\Delta =\left|\begin{array}{ccc}0& 0& 1\\ 1-a& b-1& 1\\ 1-ac& 1-c& c\end{array}\right|[C_1\to C_1-a{C}_3;C_2\to C_2-C_3]=(1-a)(1-c)-(b-1)(1-ac)=1-a-c+ac-b+abc+1-ac=(2+abc)-(a+b+c),\text{as Det}>0i,e.abc+2>a+b+c\{\because a+b+c>3(abc{)}^{1/3}\}abc+2>3(abc{)}^{1/3}\Rightarrow x^3-3x+2>0,(Letx=\left(abc{)}^{\frac{1}{3}}\right)\Rightarrow (x-1{)}^2(x+2)>0i.e.x>-2i.e.x^3=abc>-8$