Question

The roots of the equation $\left| \begin{matrix} x - 1 & 1 & 1 \\ 1 & x - 1 & 1 \\ 1 & 1 & x - 1 \end{matrix} \right| = 0$are.

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

Answer 1

Answer: (B)

– 1, 2

Answer 2

We have

$$\left| \begin{matrix} x - 1 & 1 & 1 \\ 1 & x - 1 & 1 \\ 1 & 1 & x - 1 \end{matrix} \right| = 0$$ $$\Rightarrow \left| \begin{matrix} x + 1 & 1 & 1 \\ x + 1 & x - 1 & 1 \\ x + 1 & 1 & x - 1 \end{matrix} \right| = 0,$$

{Applying $C_{1} \rightarrow C_{1} + C_{2} + C_{3}$}

$$\Rightarrow (x + 1)\left| \begin{matrix} 1 & 1 & 1 \\ 1 & x - 1 & 1 \\ 1 & 1 & x - 1 \end{matrix} \right| = 0$$ $$\Rightarrow (x + 1)\left| \begin{matrix} 1 & 1 & 1 \\ 0 & x - 2 & 0 \\ 0 & 0 & x - 2 \end{matrix} \right| = 0,$$

{Applying $R_{2} \rightarrow R_{2} - R_{1}, R_{3} \rightarrow R_{3} - R_{1}$}

$$\Rightarrow N = \begin{bmatrix} -4 & -3 & -3 \\ 1 & 0 & 1 \\ 4 & 4 & 3 \end{bmatrix}.$$