Question

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

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

Answer 1

Answer: (C)

abc

Answer 2

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