Question

The roots of the equation $\left|\begin{array}{ccc}1+x& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+x\end{array}\right|=0$ are :

• A. $0,-3$
• B. $0,1,-3$
• C. $1,-3$
• D. None of these

Answer 1

The correct option is A $0,-3$

$\begin{array}{c}\left|\begin{array}{ccc}1+x& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+x\end{array}\right|=0\\ C_1\to C_1+C_2+C_3\\ \left|\begin{array}{ccc}3+x& 1& 1\\ 3+x& 1+x& 1\\ 3+x& 1& 1+x\end{array}\right|=0\\ \left(3+x\right)\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+x\end{array}\right|=0\\ \left(3+x\right)\left|\begin{array}{ccc}1& 1& 1\\ 0& x& 0\\ 1& 1& 1+x\end{array}\right|=0\left(R_2\to R_2-R_1\right)\\ \left(3+x\right)\left(x^2\right)=0\\ \Rightarrow x=0,-3\end{array}$

Hence,

Option $A$ is correct answer.