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