Question

If $\left|\begin{array}{ccc}x+a& b& c\\ a& x+b& c\\ a& b& x+c\end{array}\right|=0$,then $x$ equals

• A. $a+b+c$
• B. $-(a+b+c)$
• C. $0,a+b+c$
• D. $0,-(a+b+c)$

Answer 1

The correct option is D $0,-(a+b+c)$

Given, $\left|\begin{array}{ccc}x+a& b& c\\ a& x+b& c\\ a& b& x+c\end{array}\right|=0$
$C_1\to C_1+C_2+C_3$
$\left|\begin{array}{ccc}x+a+b+c& b& c\\ x+a+b+c& x+b& c\\ x+a+b+c& b& x+c\end{array}\right|=0$
$(x+a+b+c)\left|\begin{array}{ccc}1& b& c\\ 1& x+b& c\\ 1& b& x+c\end{array}\right|=0$
$R_1\to R_1-R_2,R_2\to R_2-R_3$
$(x+a+b+c)\left|\begin{array}{ccc}0& -x& 0\\ 0& x& -x\\ 1& b& x+c\end{array}\right|=0$
$\Rightarrow x(x+a+b+c)\left|\begin{array}{ccc}0& -x& 0\\ 0& 1& -1\\ 1& b& x+c\end{array}\right|=0$
$\Rightarrow x^2(x+a+b+c)=0$
$\Rightarrow x=0,-(a+b+c)$