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

Answer 1

The correct option is A 0 and $-(\alpha +\beta +\gamma )$

Equation given,$\left|\begin{array}{ccc}x+\alpha +\beta +\gamma & \beta & \gamma \\ x+\alpha +\beta +\gamma & x+\beta & \alpha \\ x+\alpha +\beta +\gamma & \beta & x+\gamma \end{array}\right|=0$
$c_1\to c_1+c_2+c_3$
or $(x+\alpha +\beta +\gamma )\left|\begin{array}{ccc}1& \beta & \gamma \\ 1& x+\beta & \alpha \\ 1& \beta & x+\gamma \end{array}\right|=0$
or $(x+\alpha +\beta +\gamma )\left|\begin{array}{ccc}1& \beta & \gamma \\ 0& x& \alpha -\gamma \\ 0& 0& x\end{array}\right|=0,\left[\begin{array}{cc}& R_2\to R_2-R_1\\ & R_3\to R_3-R_1\end{array}\right]$
or $(x+\alpha +\beta +\gamma )[x^2-0]=0$
or $x^2(x+\alpha +\beta +\gamma )=0$
$\therefore x=0orx=-(\alpha +\beta +\gamma ).$