Question

One of the roots of the given equation $\left|\begin{array}{ccc}x+a& b& c\\ b& x+c& a\\ c& a& x+b\end{array}\right|=0$ is

• A. -(a+b)
• B. -(b+c)
• C. -a
• D. -(a+b+c)

Answer 1

The correct option is D -(a+b+c)

$\left|\begin{array}{ccc}x+a& b& c\\ b& x+c& a\\ c& a& x+b\end{array}\right|=0\Rightarrow (x+a+b+c)\left|\begin{array}{ccc}1& b& c\\ 1& x+c& a\\ 1& a& x+b\end{array}\right|=0,(C_1\to C_1+C_2+C_3)\Rightarrow x=-(a+b+c)$ is one of the root of the equation.