Question

If a,b,c are non – zero real numbers and if the equations (a-1) x = y + z, (b -1)y = z + x, (c - 1)z = x + y has a non trivial solution, then ab + bc + ca is equal to

• A. a +b + c
• B. abc
• C. 1
• D. None of these

Answer 1

The correct option is B abc

For non trivial solution
$\left|\begin{array}{ccc}a-1& -1& -1\\ 1& 1-b& 1\\ 1& 1& 1-c\end{array}\right|=0$
Applying $C_1\to C_1-C_3$ and $C_2\to C_2-C_3,$ then
$\left|\begin{array}{ccc}a& 0& 1\\ 0& -b& 1\\ c& c& 1-c\end{array}\right|=0\Rightarrow a(-b+bc-c)-0+c(0-b)=0\Rightarrow ab+bc+ca=abc$