Question

If the system of equations ax + y + z = 0, x + by + z = 0 and x+y+cz = 0, where $a,b,c\ne 1$, has a nontrivial solution, then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is

• A. -1
• B. 1
• C. None of these

Answer 1

The correct option is B 1

As the system of equations has a non-trivial solution
$\Rightarrow \left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|=0\Rightarrow \left|\begin{array}{ccc}a& 1& 1\\ 1-a& b-1& 0\\ 1-a& 0& c-1\end{array}\right|=0,by\begin{array}{c}{R}_2\to R_2-R_1\\ R_3\to R_3-R_1\end{array}\Rightarrow a(b-1)(c-1)-1.(1-a)(c-1)-1.(1-a)(b-1)=0\Rightarrow \frac{a}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=0\Rightarrow \frac{1}{1-a}-1+\frac{1}{1-b}+\frac{1}{1-c}=0\Rightarrow \frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=1.$