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Question: If \(a^{- 1} + b^{- 1} + c^{- 1} = 0\) such that\(\left| \begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b &...

If a−1+b−1+c−1=0a^{- 1} + b^{- 1} + c^{- 1} = 0 such that∣1+a1111+b1111+c∣=λ\left| \begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{matrix} \right| = \lambda, then the value of λ\lambdais.

A

0

B

abc

C

– abc

D

None of these

Answer

abc

Explanation

Solution

k=1,∣A∣=0\mathbf{k = 1,|A| = 0}

Applying $5A = \begin{bmatrix} 15 & - 25 \

  • 20 & 10 \end{bmatrix}andandC_{3} \rightarrow C_{3} - C_{1},$
1 + a & - a & - a \\ 1 & b & 0 \\ 1 & 0 & c \end{matrix} \right|$$ On expanding w.r.t. $R_{3}$, $ab + bc + ca + abc = \lambda$ …….(i) Given, $a^{- 1} + b^{- 1} + c^{- 1} = 0$ ⇒ $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 0$ ⇒ $ab + bc + ca = 0$ ⇒ $\lambda = abc$, (From equation (i)).