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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 non trivial solution, then the value of $\frac{1}{{1 - a}} + \frac{1}{{1 - b}} + \frac{1}{{1 - c}}$is
$-1$
$0$
$1$
None of these
Solution
(c) As the system of equations has a non-trivial solution
$ \Rightarrow $ $\left| {\,\begin{array}{*{20}{c}}a&1&1\\1&b&1\\1&1&c\end{array}\,} \right|\, = \,0$
$ \Rightarrow $ $\left| {\,\begin{array}{*{20}{c}}a&1&1\\{1 – a}&{b – 1}&0\\{1 – a}&0&{c – 1}\end{array}\,} \right|\, = 0,$
by $\begin{array}{l}{R_2} \to {R_2} – {R_1}\\{R_3} \to {R_3} – {R_1}\end{array}$
$ \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 $ $\frac{1}{{1 – a}} + \frac{1}{{1 – b}} + \frac{1}{{1 – c}} = 1$.
