Question

If $a^{-1}+b^{-1}+c^{-1}=0$ such that
$\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=\lambda$ then the value of $\lambda$ is :

• A. 0
• B. - abc
• C. abc
• D. None of these

Answer 1

Answer: (C)

Given: $a^{-1}+b^{-1}+c^{-1}=0$
and $\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=\lambda$,
$\Rightarrow abc\left|\begin{array}{ccc}\frac{1+a}{a}& \frac{1}{a}& \frac{1}{a}\\ \frac{1}{b}& \frac{1+b}{b}& \frac{1}{b}\\ \frac{1}{c}& \frac{1}{c}& \frac{1+c}{c}\end{array}\right|=\lambda$
$\Rightarrow abc\left|\begin{array}{ccc}\frac{1}{a}+1& \frac{1}{a}& \frac{1}{a}\\ \frac{1}{b}& \frac{1}{b}+1& \frac{1}{b}\\ \frac{1}{c}& \frac{1}{c}& \frac{1}{c}+1\end{array}\right|=\lambda$
$\left(R_1\to R_1+R_2+R_3\right)$
$\Rightarrow abc\left|\begin{array}{ccc}\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1& \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1& \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1\\ \frac{1}{b}& \frac{1}{b}+1& \frac{1}{b}\\ \frac{1}{c}& \frac{1}{c}& \frac{1}{c}+1\end{array}\right|=\lambda$
$\Rightarrow abc(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1)\left|\begin{array}{ccc}1& 1& 1\\ \frac{1}{b}& \frac{1}{b}+1& \frac{1}{b}\\ \frac{1}{c}& \frac{1}{c}& \frac{1}{c}+1\end{array}\right|=\lambda$
$\Rightarrow abc\left|\begin{array}{ccc}1& 1& 1\\ \frac{1}{b}& \frac{1}{b}+1& \frac{1}{b}\\ \frac{1}{c}& \frac{1}{c}& \frac{1}{c}+1\end{array}\right|=\lambda$
$R_2\to \frac{1}{b}{R}_1-R_2,R_3\to \frac{1}{c}{R}_1-R_3$
$\Rightarrow abc\left|\begin{array}{ccc}1& 1& 1\\ 0& -1& 0\\ 0& 0& -1\end{array}\right|=\lambda$
$\Rightarrow abc=\lambda$