Question

The roots of the equation $\begin{vmatrix}x-1&1&1\\ 1&x-1&1\\ 1&1&x-1\end{vmatrix} = 0 $ are

• A. 1,2
• B. -1,2
• C. -1,-2
• D. 1,-2

Answer 1

The Correct Option is B

We have, $\begin{vmatrix}x-1 & 1 & 1 \\ 1 & x-1 & 1 \\ 1 & 1 & x-1\end{vmatrix}=0$ On applying $C_{1} \rightarrow C_{1}+C_{2}+C_{3}$, we get $\begin{vmatrix}x+1 & 1 & 1 \\ x+1 & x-1 & 1 \\ x+1 & 1 & x-1\end{vmatrix}=0$ On taking $(x+1)$ common from $C_{1}$, we get $(x+1)\begin{vmatrix}1 & 1 & 1 \\ 1 & x-1 & 1 \\ 1 & 1 & x-1\end{vmatrix}=0$ On applying, $R_{1} \rightarrow R_{1}-R_{2}, R_{2} \rightarrow R_{2}-R_{3}$, we get $\Rightarrow(x+1)\begin{vmatrix}0 & 2-x & 0 \\ 0 & x-2 & 2-x \\ 1 & 1 & x-1\end{vmatrix}=0$ $\Rightarrow (x+1) .1\left[(2-x)^{2}-0\right]=0$ $\Rightarrow (x+1)(2-x)^{2}=0$ $\Rightarrow x=-1,2$