Question

One of the roots of the given equation

$\left| \begin{matrix} x + a & b & c \\ b & x + c & a \\ c & a & x + b \end{matrix} \right| = 0$ is.

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

Answer 1

Answer: (D)

$- (a + b + c)$

Answer 2

$\left| \begin{matrix} x + a & b & c \\ b & x + c & a \\ c & a & x + b \end{matrix} \right| = 0$

$\Rightarrow (x + a + b + c)\left| \begin{matrix} 1 & b & c \\ 1 & x + c & a \\ 1 & a & x + b \end{matrix} \right| = 0$ ,

$= a\{ a^{2} + ab - 2a^{2} - ab\rbrack = - a^{3} = i$

$\Rightarrow x = - (a + b + c)$ is one of the root of the equation.