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 $ab + bc + ca$ equals.

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

Answer 1

Answer: (B)

abc

Answer 2

For non-trivial solution

$\left| \begin{matrix} a - 1 & - 1 & - 1 \\ 1 & 1 - b & 1 \\ 1 & 1 & 1 - c \end{matrix} \right| = 0$

Applying $C_{1} \rightarrow C_{1} - C_{2}andC_{2} \rightarrow C_{2} - C_{3}$then $\left| \begin{matrix} a & 0 & - 1 \\ b & - b & 1 \\ 0 & c & 1 - c \end{matrix} \right| = 0$

⇒ $a( - b + bc - c) - 0 - 1(bc + 0) = 0$

⇒ $- ab + abc - ac - bc = 0$

∴ $ab + bc + ca = abc$