Question

If the system of equations $ax + y + z = 0,x + by + z = 0$ and

$x + y + cz = 0$, where $a,b,c \neq 1$ has a non-trivial solution, then the value of $\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c}$ is

• A. –1
• B. 0
• C. 1
• D. None of these

Answer 1

Answer: (C)

1

Answer 2

As the system of the equations has a non-trivial solution $\left| \begin{matrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{matrix} \right| = 0$

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

a & 1 & 1 \\ 1 - a & b - 1 & 0 \\ 1 - a & 0 & c - 1 \end{matrix} \right| = 0$$ $$\Rightarrow a(b - 1)(c - 1) - 1(1 - a)(c - 1) - 1(1 - a)(b - 1) = 0$$ $\Rightarrow$ $\frac{a}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} = 0$ $\Rightarrow$ $\frac{1}{1 - a} - 1 + \frac{1}{1 - b} + \frac{1}{1 - c} = 0$ $\Rightarrow$ $\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} = 1$