Question

If a, b, c are non-zero real numbers and if the system of 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. -1

Answer 1

Answer: (B)

for non-trivial solution $D =0$
$\begin{vmatrix}1- a & 1 & 1 \\ 1 & 1- b & 1 \\ 1 & 1 & 1- c \end{vmatrix}=0$
$R _{1} \rightarrow R _{1}- R _{3} $
$\begin{vmatrix}- a & 0 & c \\ 1 & 1- b & 1 \\ 1 & 1 & 1- c \end{vmatrix}$
$\Rightarrow $ a $\{(1-b)(1-c)-1\}+c\{1-(1-b)\}=0$
$\Rightarrow a b+a c+b c-a b c=0$
$\Rightarrow a b+a c+b c=a b c$