If a, b, c are non-zero and different from 1, then the value of is

Engineering

Mathematics

Introduction to Determinants

Question

If a, b, c are non-zero and different from 1, then the value of $∣ \begin{array}{ccc} \log_{a} 1 & \log_{a} b & \log_{a} c \\ \log_{a} (\frac{1}{b}) & \log_{b} 1 & \log_{a} (\frac{1}{c}) \\ \log_{a} (\frac{1}{c}) & \log_{a} c & \log_{c} 1 \end{array} ∣$ is

log_a(ab + bc + ca)

log_a(a + b + c)

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$Δ = ∣ \begin{array}{ccc} \log_{a} 1 & \log_{a} b & \log_{a} c \\ \log_{a} \frac{1}{b} & \log_{b} 1 & \log_{a} \frac{1}{c} \\ \log_{a} \frac{1}{c} & \log_{a} c & \log_{c} 1 \end{array} ∣$

$\Rightarrow Δ = ∣ \begin{array}{ccc} 0 & \log_{a} b & \log_{a} c \\ - \log_{a} b & 0 & - \log_{a} c \\ - \log_{a} c & \log_{a} c & 0 \end{array} ∣$

⇒ Δ = 0 – log_ab {0 – (log_ac)²} + log_ac {– log_ab log_ac – 0}

⇒ Δ = log_ab(log_ac)² – log_ab(log_ac)² = 0

So, option A is correct.

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