Question

The value of $x$ obtained from the equation $\left|\begin{array}{ccc}x+\alpha & \beta & \gamma \\ \gamma & x+\beta & \alpha \\ \alpha & \beta & x+\gamma \end{array}\right|=0$ will be

• A. $0$ and $-\left(\alpha +\beta +\gamma \right)$
• B. $0$ and $\alpha +\beta +\gamma$
• C. $1$ and $\left(\alpha -\beta -\gamma \right)$
• D. $0$ and ${\alpha }^2+\beta +{\gamma }^2$

Answer 1

Answer: (A)

Given $\left|\begin{array}{ccc}x+\alpha & \beta & \gamma \\ \gamma & x+\beta & \alpha \\ \alpha & \beta & x+\gamma \end{array}\right|=0$
Operate $C_1\to C_1+C_2+C_3$
$\left|\begin{array}{ccc}x+\alpha +\beta +\gamma & \beta & \gamma \\ x+\alpha +\beta +\gamma & x+\beta & \alpha \\ x+\alpha +\beta +\gamma & \beta & x+\gamma \end{array}\right|=0$
$=\left(x+\alpha +\beta +\gamma \right)\left|\begin{array}{ccc}1& \beta & \gamma \\ 1& x+\beta & \alpha \\ 1& \beta & x+\gamma \end{array}\right|=0$
$\Rightarrow x+\alpha +\beta +\gamma =0\Rightarrow x=-\left(\alpha +\beta +\gamma \right)$ Again if
$\left|\begin{array}{ccc}1& \beta & \gamma \\ 1& x+\beta & \alpha \\ 1& \beta & \gamma \end{array}\right|=0\Rightarrow \left|\begin{array}{ccc}1& \beta & \gamma \\ 0& x& \alpha -\gamma \\ 0& 0& x\end{array}\right|=0$
$\Rightarrow x^2=0\Rightarrow x=0$
$\therefore$ Solutions of the equation are $x=0$,
$-\left(\alpha +\beta +\gamma \right)$