Question

If $a+b+c=0$, one root of $\left|\begin{array}{ccc}a-x& c& b\\ c& b-x& a\\ b& a& c-x\end{array}\right|=0$ is

• A. $x=1$
• B. $x=2$
• C. $x=a^2+b^2+c$
• D. $x=0$

Answer 1

The correct option is D $x=0$

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

$R_1\to R_1-R_2,R_2-\to R_2-R_3$

$=-x\left|\begin{array}{ccc}0& c-b+x& b-a\\ 0& b-x-a& a-c+x\\ 1& a& c-x\end{array}\right|=0$
On expanding above,we get
$\Rightarrow x=0$