Question

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

• A. 0, – 3
• B. 0, 0, – 3
• C. 0, 0, 0, – 3
• D. None of these

Answer 1

Answer: (B)

0, 0, – 3

Answer 2

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

$\Rightarrow 4\left| \begin{matrix} 1 & 21 & 9 \\ 0 & 90 & - 45 \\ 0 & - 4 & 2 \end{matrix} \right|$, $\left( \begin{aligned} & C_{1} \rightarrow C_{1} + C_{2} + C_{3} \\ & C_{2} \rightarrow C_{2} - C_{3} \end{aligned} \right)$

1 & 0 & 1 \\ 1 & x & 1 \\ 1 & - x & 1 + x \end{matrix} \right| = 0$$ $\Rightarrow (x + 3)\left| \begin{matrix} 1 & 0 & 1 \\ 0 & x & 0 \\ 0 & - x & x \end{matrix} \right| = 0$, $\left( \begin{aligned} & R_{2} \rightarrow R_{2} - R_{1} \\ & R_{3} \rightarrow R_{3} - R_{1} \end{aligned} \right)$ $\Rightarrow (x + 3)x^{2} = 0 \Rightarrow x = 0,0, - 3$. **Trick :** Obviously the equation is of degree three, therefore it must have three solutions. So check for option (2).