Question

If $a,b,c$ are roots of the equation $A{x}^3+B{x}^2+C=0$ then the value of determinant $\left|\begin{array}{ccc}ab& bc& ca\\ bc& ca& ab\\ ca& ab& bc\end{array}\right|$ equals to=?

• A. $A$
• B. $B$
• C. $C$
• D. $0$

Answer 1

The correct option is D $0$

Given : $a,b,c$ are roots of the equation $A{x}^3+B{x}^2+C=0$
$\Rightarrow ab+bc+ac=0$
$\left|\begin{array}{ccc}ab& bc& ca\\ bc& ca& ab\\ ca& ab& bc\end{array}\right|$
$C_1\to C_1+C_2+C_3$
$=\left|\begin{array}{ccc}ab+bc+ac& bc& ca\\ ab+bc+ac& ca& ab\\ ab+bc+ac& ab& bc\end{array}\right|$
$=\left|\begin{array}{ccc}0& bc& ca\\ 0& ca& ab\\ 0& ab& bc\end{array}\right|$
$=0$