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  4. If a-1 + b-1 + c-1 = 0 such that |1+a&1&1 1&1+b&1 1&1&1+c| = λ then the value of λ is :

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Q. 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 :

Determinants

Solution:

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$