Question

One of the roots of $\left|\begin{array}{ccc}x+a& b& c\\ a& x+b& c\\ a& b& x+c\end{array}\right|=0$ is

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

Answer 1

The correct option is C $-(a+b+c)$

Given,

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

[$C_1^{\prime }=C_1+C_2+C_3$]

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

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

[$R_2^{\prime }=R_2-R_1$ and $R_3^{\prime }=R_3-R_1$]

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

or, $x^2(x+a+b+c)=0$

So the roots are $0.0,-(a+b+c)$.

One root is $-(a+b+c)$.